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Kalkulator do pochodnych

Krystian Karczyński

Założyciel i szef serwisu eTrapez.

Magister matematyki Politechniki Poznańskiej. Korepetytor matematyki z wieloletnim stażem. Twórca Kursów eTrapez, które zdobyły ogromną popularność wśród studentów w całej Polsce.

Mieszka koło Szczecina. Lubi spacery po lesie, plażowanie i piłkę nożną.


Zapraszam do korzystania z przerobionego przeze mnie kalkulatora do pochodnych: Myślę, że tutaj sprawa jest bardzo jasna. Wpisujemy funkcję, klikamy na ‘Oblicz” i mamy jej pochodną. Funkcje należy wpisywać we właściwy sposób, zgodny z ogólną instrukcją wpisywania formuł matematycznych. Poniżej kilka przykładów.

Przykład 1

Chcemy obliczyć pochodną z funkcji y equals 4 x cubed. Wpisujemy w kalkulator: 4x^3. Klikamy ‘Oblicz’. Mamy wynik: y apostrophe equals 12 x squared

Przykład 2

Chcemy obliczyć pochodną z funkcji y equals ln squared open parentheses sin x plus 12 close parentheses. Wpisujemy w kalkulator: (ln(sinx+12))^2 Mamy wynik: y apostrophe equals fraction numerator 2 cos x ln open parentheses sin x plus 12 close parentheses over denominator sin x plus 12 end fraction

Przykład 3

Chcemy obliczyć pochodną z funkcji y equals fraction numerator x plus 1 over denominator open parentheses x minus 2 close parentheses open parentheses x plus 4 close parentheses end fraction. Wpisujemy w kalkulator: (x+1)/((x-2)(x+4)) Mamy wynik: y apostrophe equals horizontal ellipsis sami sprawdźcie jaki (trochę kosmiczny, ale tylko trochę) 🙂

Jedna z wielu opinii o naszych Kursach...

Nigdy nie byłem dobry z matmy a wybrałem się na studia inżynierskie, teraz zaczynam wszystko rozumieć. Jednak nauczyciel ma znaczenie!

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  1. Klaudia pisze:

    Dzień dobry, jak wprowadzic pierwiastek w kalkulator aby obejmował całe wyrażenie a nie tylko daną część?

  2. Antytalencię pisze:

    Dzień dobry.Nwm jak policzyć  pochodną f(x) =sin(2 do x).Wię piszę tutaj

  3. Witam, polecam moją darmową Lekcję do liczenia pochodnej z definicji 🙂

    A co przykładu, poleci tak:

    Wzór na pochodną w punkcie x subscript 0 z definicji to:

    f apostrophe open parentheses x subscript 0 close parentheses equals limit as increment x rightwards arrow 0 of fraction numerator f open parentheses x subscript 0 plus increment x close parentheses minus f open parentheses x subscript 0 close parentheses over denominator increment x end fraction

    W naszym przypadku f open parentheses x close parentheses equals fraction numerator 1 over denominator 5 x plus 6 end fraction .

    Mamy więc:

    f apostrophe open parentheses x subscript 0 close parentheses equals limit as increment x rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator 5 open parentheses x subscript 0 plus increment x close parentheses plus 6 end fraction end style minus begin display style fraction numerator 1 over denominator 5 x subscript 0 plus 6 end fraction end style over denominator increment x end fraction equals limit as increment x rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator 5 x subscript 0 plus 5 increment x plus 6 end fraction minus fraction numerator 1 over denominator 5 x subscript 0 plus 6 end fraction end style over denominator increment x end fraction equals
equals limit as increment x rightwards arrow 0 of fraction numerator begin display style fraction numerator 5 x subscript 0 plus 6 over denominator open parentheses 5 x subscript 0 plus 5 increment x plus 6 close parentheses open parentheses 5 x subscript 0 plus 6 close parentheses end fraction minus fraction numerator 5 x subscript 0 plus 5 increment x plus 6 over denominator open parentheses 5 x subscript 0 plus 5 increment x plus 6 close parentheses open parentheses 5 x subscript 0 plus 6 close parentheses end fraction end style over denominator increment x end fraction equals

    equals limit as increment x rightwards arrow 0 of fraction numerator begin display style fraction numerator 5 x subscript 0 plus 6 minus open parentheses 5 x subscript 0 plus 5 increment x plus 6 close parentheses over denominator open parentheses 5 x subscript 0 plus 5 increment x plus 6 close parentheses open parentheses 5 x subscript 0 plus 6 close parentheses end fraction end style over denominator increment x end fraction equals limit as increment x rightwards arrow 0 of fraction numerator begin display style fraction numerator 5 x subscript 0 plus 6 minus 5 x subscript 0 minus 5 increment x minus 6 over denominator open parentheses 5 x subscript 0 plus 5 increment x plus 6 close parentheses open parentheses 5 x subscript 0 plus 6 close parentheses end fraction end style over denominator increment x end fraction equals
equals limit as increment x rightwards arrow 0 of fraction numerator begin display style fraction numerator negative 5 increment x over denominator open parentheses 5 x subscript 0 plus 5 increment x plus 6 close parentheses open parentheses 5 x subscript 0 plus 6 close parentheses end fraction end style over denominator increment x end fraction equals limit as increment x rightwards arrow 0 of fraction numerator negative 5 increment x over denominator open parentheses 5 x subscript 0 plus 5 increment x plus 6 close parentheses open parentheses 5 x subscript 0 plus 6 close parentheses end fraction fraction numerator begin display style 1 end style over denominator increment x end fraction equals
equals limit as increment x rightwards arrow 0 of fraction numerator negative 5 over denominator open parentheses 5 x subscript 0 plus 5 increment x plus 6 close parentheses open parentheses 5 x subscript 0 plus 6 close parentheses end fraction equals fraction numerator negative 5 over denominator open parentheses 5 x subscript 0 plus 6 close parentheses open parentheses 5 x subscript 0 plus 6 close parentheses end fraction equals fraction numerator negative 5 over denominator open parentheses 5 x subscript 0 plus 6 close parentheses squared end fraction

    Sprawdzamy prawdziwość tego wyniku korzystając ze wzorów:

    f apostrophe open parentheses x subscript 0 close parentheses equals open parentheses fraction numerator 1 over denominator 5 x subscript 0 plus 6 end fraction close parentheses apostrophe equals open square brackets open parentheses 5 x subscript 0 plus 6 close parentheses to the power of negative 1 end exponent close square brackets apostrophe equals negative 1 times open parentheses 5 x subscript 0 plus 6 close parentheses to the power of negative 1 minus 1 end exponent open parentheses 5 x subscript 0 plus 6 close parentheses apostrophe equals
equals negative 1 open parentheses 5 x subscript 0 plus 6 close parentheses to the power of negative 2 end exponent times 5 equals fraction numerator negative 5 over denominator open parentheses 5 x subscript 0 plus 6 close parentheses squared end fraction

    Czyli wszystko gra 🙂

     

    74/a0/d9bf2a1d5480ed8a61a2e0d8df7b.png” alt=”open parentheses co ś close parentheses to the power of n” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mfenced»«mrow»«mi»co«/mi»«mi»§#347;«/mi»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«/math»” /> . Przy liczeniu pochodnej wykorzystujesz wzór na open parentheses x to the power of n close parentheses apostrophe equals n times x to the power of n minus 1 end exponent , z tym, że trzeba pamiętać do DOMNOŻENIU jeszcze pochodnej tego czegoś więcej, tego wyrażenia “coś”, czyli: open parentheses open parentheses co ś close parentheses to the power of n space close parentheses apostrophe equals space n times open parentheses co ś close parentheses to the power of n minus 1 end exponent times open parentheses c o ś close parentheses apostrophe

    No tu wyjdzie ostatecznie:

    open square brackets fraction numerator 2 x to the power of 6 minus 16 x cubed over denominator open parentheses x cubed minus 2 close parentheses squared end fraction close square brackets apostrophe equals fraction numerator open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses apostrophe times space open parentheses x cubed minus 2 close parentheses squared space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space open square brackets open parentheses x cubed minus 2 close parentheses squared close square brackets apostrophe over denominator open square brackets open parentheses x cubed minus 2 close parentheses squared close square brackets squared end fraction equals

    equals fraction numerator open parentheses 2 times 6 times x to the power of 5 minus 16 times 3 times x squared close parentheses times space open parentheses x cubed minus 2 close parentheses squared space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space open square brackets 2 times open parentheses x cubed minus 2 close parentheses to the power of 1 times open parentheses x cubed minus 2 close parentheses apostrophe close square brackets over denominator open parentheses x cubed minus 2 close parentheses to the power of 4 end fraction equals

    equals fraction numerator open parentheses 12 x to the power of 5 minus 48 x squared close parentheses times open parentheses x cubed minus 2 close parentheses squared space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space open square brackets 2 open parentheses x cubed minus 2 close parentheses times open parentheses 3 times x squared minus 0 close parentheses close square brackets over denominator open parentheses x cubed minus 2 close parentheses to the power of 4 end fraction equals

    equals fraction numerator open parentheses 12 x to the power of 5 minus 48 x squared close parentheses times open parentheses x cubed minus 2 close parentheses squared space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space 6 x squared open parentheses x cubed minus 2 close parentheses over denominator open parentheses x cubed minus 2 close parentheses to the power of 4 end fraction equals

    equals fraction numerator open parentheses x cubed minus 2 close parentheses open square brackets open parentheses 12 x to the power of 5 minus 48 x squared close parentheses times open parentheses x cubed minus 2 close parentheses space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space 6 x squared close square brackets over denominator open parentheses x cubed minus 2 close parentheses to the power of 4 end fraction equals

    equals fraction numerator 12 x to the power of 8 minus 24 x to the power of 5 minus 48 x to the power of 5 plus 96 x squared minus space 12 x to the power of 8 plus 96 x to the power of 5 over denominator open parentheses x cubed minus 2 close parentheses cubed end fraction equals fraction numerator 24 x to the power of 5 plus 96 x squared over denominator open parentheses x cubed minus 2 close parentheses cubed end fraction equals fraction numerator bold 24 bold x to the power of bold 2 open parentheses bold x to the power of bold 3 bold plus bold 4 close parentheses over denominator open parentheses bold x to the power of bold 3 bold minus bold 2 close parentheses to the power of bold 3 end fraction

  4. Magda pisze:

    Witam,

    mam problem z rozwiązaniem takiego zadania:

    Oblicz z definicji pochodną f(x)= 1/(5x+6) w punkcie x0. Poprawność sprawdź z wzorów na pochodne.

    Z góry dziękuję za pomoc.

  5. Natalia pisze:

    Panie Krystianie, nie do końca wiem jak obliczyć pochodną z funkcji (2x^6-16x^3)/(x^3-2)^2. Mógłby Pan mi prosze pomóc? 🙂

    1. Pochodna z fraction numerator 2 x to the power of 6 minus 16 x cubed over denominator open parentheses x cubed minus 2 close parentheses squared end fraction.

      Na początku mamy tutaj dzielenie dwóch funkcji, więc zaczynamy od zastosowania wzoru: open parentheses f over g close parentheses apostrophe equals fraction numerator f apostrophe space times g space minus space f times g apostrophe over denominator g squared end fraction

      f equals 2 x to the power of 6 minus 16 x cubed – tutaj spoko, licząc pochodną wykorzystujemy liniowość, czyli pochodna z każdego składnika oddzielnie oraz dwa proste wzory: open square brackets a times f left parenthesis x right parenthesis close square brackets apostrophe equals a times open square brackets f left parenthesis x right parenthesis close square brackets apostrophe (stała przed pochodną po x-sie), a także wzór:   open parentheses x to the power of n close parentheses apostrophe equals n times x to the power of n minus 1 end exponent.

      g equals open parentheses x cubed minus 2 close parentheses squared – tutaj występuje takie coś jak złożenie funkcji. Masz jakieś wyrażenie podniesione do potęgi drugiej, czyli open parentheses co ś close parentheses to the power of n . Przy liczeniu pochodnej wykorzystujesz wzór na open parentheses x to the power of n close parentheses apostrophe equals n times x to the power of n minus 1 end exponent , z tym, że trzeba pamiętać do DOMNOŻENIU jeszcze pochodnej tego czegoś więcej, tego wyrażenia “coś”, czyli: open parentheses open parentheses co ś close parentheses to the power of n space close parentheses apostrophe equals space n times open parentheses co ś close parentheses to the power of n minus 1 end exponent times open parentheses c o ś close parentheses apostrophe

      No tu wyjdzie ostatecznie:

      open square brackets fraction numerator 2 x to the power of 6 minus 16 x cubed over denominator open parentheses x cubed minus 2 close parentheses squared end fraction close square brackets apostrophe equals fraction numerator open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses apostrophe times space open parentheses x cubed minus 2 close parentheses squared space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space open square brackets open parentheses x cubed minus 2 close parentheses squared close square brackets apostrophe over denominator open square brackets open parentheses x cubed minus 2 close parentheses squared close square brackets squared end fraction equals

      equals fraction numerator open parentheses 2 times 6 times x to the power of 5 minus 16 times 3 times x squared close parentheses times space open parentheses x cubed minus 2 close parentheses squared space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space open square brackets 2 times open parentheses x cubed minus 2 close parentheses to the power of 1 times open parentheses x cubed minus 2 close parentheses apostrophe close square brackets over denominator open parentheses x cubed minus 2 close parentheses to the power of 4 end fraction equals

      equals fraction numerator open parentheses 12 x to the power of 5 minus 48 x squared close parentheses times open parentheses x cubed minus 2 close parentheses squared space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space open square brackets 2 open parentheses x cubed minus 2 close parentheses times open parentheses 3 times x squared minus 0 close parentheses close square brackets over denominator open parentheses x cubed minus 2 close parentheses to the power of 4 end fraction equals

      equals fraction numerator open parentheses 12 x to the power of 5 minus 48 x squared close parentheses times open parentheses x cubed minus 2 close parentheses squared space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space 6 x squared open parentheses x cubed minus 2 close parentheses over denominator open parentheses x cubed minus 2 close parentheses to the power of 4 end fraction equals

      equals fraction numerator open parentheses x cubed minus 2 close parentheses open square brackets open parentheses 12 x to the power of 5 minus 48 x squared close parentheses times open parentheses x cubed minus 2 close parentheses space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space 6 x squared close square brackets over denominator open parentheses x cubed minus 2 close parentheses to the power of 4 end fraction equals

      equals fraction numerator 12 x to the power of 8 minus 24 x to the power of 5 minus 48 x to the power of 5 plus 96 x squared minus space 12 x to the power of 8 plus 96 x to the power of 5 over denominator open parentheses x cubed minus 2 close parentheses cubed end fraction equals fraction numerator 24 x to the power of 5 plus 96 x squared over denominator open parentheses x cubed minus 2 close parentheses cubed end fraction equals fraction numerator bold 24 bold x to the power of bold 2 open parentheses bold x to the power of bold 3 bold plus bold 4 close parentheses over denominator open parentheses bold x to the power of bold 3 bold minus bold 2 close parentheses to the power of bold 3 end fraction

  6. damian pisze:

    x^2+e^x/x-lnx czy pomoze ktoś ?

  7. Kamila pisze:

     WitamMam problem z pochodna x^x jak to obliczyc?

    1. Myślę, że ten filmik będzie baaardzo pomocny i wszystko wyjaśniający (chociaż przykład jest lekko inny) 🙂

       

      W Pani przypadku wyjdzie ostatecznie:

      open parentheses x to the power of x close parentheses apostrophe space equals open parentheses e to the power of ln x to the power of x end exponent close parentheses apostrophe equals space open parentheses e to the power of x times ln x end exponent close parentheses apostrophe equals open parentheses e to the power of x times ln x end exponent close parentheses times open parentheses x times ln x close parentheses apostrophe space equals

      equals open parentheses e to the power of x times ln x end exponent close parentheses times open parentheses open parentheses x close parentheses apostrophe times ln x plus x times open parentheses ln x close parentheses apostrophe close parentheses equals space open parentheses e to the power of x times ln x end exponent close parentheses times open parentheses 1 times ln x plus x times 1 over x close parentheses equals

      equals bold space straight e to the power of straight x times lnx end exponent times open parentheses lnx plus 1 close parentheses space equals space straight e to the power of lnx to the power of x end exponent times open parentheses lnx plus 1 close parentheses space equals space bold italic x to the power of bold x bold times open parentheses bold l bold n bold x bold plus bold 1 close parentheses

      8c/62/c4072d1856668341a7684910e377.png” alt=”open square brackets a times f left parenthesis x right parenthesis close square brackets apostrophe equals a times open square brackets f left parenthesis x right parenthesis close square brackets apostrophe” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»a«/mi»«mo»§#183;«/mo»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«/mfenced»«mo»`«/mo»«mo»=«/mo»«mi»a«/mi»«mo»§#183;«/mo»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«/mfenced»«mo»`«/mo»«/math»” /> (stała przed pochodną po x-sie), a także wzór:   open parentheses x to the power of n close parentheses apostrophe equals n times x to the power of n minus 1 end exponent.

      g equals open parentheses x cubed minus 2 close parentheses squared – tutaj występuje takie coś jak złożenie funkcji. Masz jakieś wyrażenie podniesione do potęgi drugiej, czyli open parentheses co ś close parentheses to the power of n . Przy liczeniu pochodnej wykorzystujesz wzór na open parentheses x to the power of n close parentheses apostrophe equals n times x to the power of n minus 1 end exponent , z tym, że trzeba pamiętać do DOMNOŻENIU jeszcze pochodnej tego czegoś więcej, tego wyrażenia “coś”, czyli: open parentheses open parentheses co ś close parentheses to the power of n space close parentheses apostrophe equals space n times open parentheses co ś close parentheses to the power of n minus 1 end exponent times open parentheses c o ś close parentheses apostrophe

      No tu wyjdzie ostatecznie:

      open square brackets fraction numerator 2 x to the power of 6 minus 16 x cubed over denominator open parentheses x cubed minus 2 close parentheses squared end fraction close square brackets apostrophe equals fraction numerator open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses apostrophe times space open parentheses x cubed minus 2 close parentheses squared space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space open square brackets open parentheses x cubed minus 2 close parentheses squared close square brackets apostrophe over denominator open square brackets open parentheses x cubed minus 2 close parentheses squared close square brackets squared end fraction equals

      equals fraction numerator open parentheses 2 times 6 times x to the power of 5 minus 16 times 3 times x squared close parentheses times space open parentheses x cubed minus 2 close parentheses squared space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space open square brackets 2 times open parentheses x cubed minus 2 close parentheses to the power of 1 times open parentheses x cubed minus 2 close parentheses apostrophe close square brackets over denominator open parentheses x cubed minus 2 close parentheses to the power of 4 end fraction equals

      equals fraction numerator open parentheses 12 x to the power of 5 minus 48 x squared close parentheses times open parentheses x cubed minus 2 close parentheses squared space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space open square brackets 2 open parentheses x cubed minus 2 close parentheses times open parentheses 3 times x squared minus 0 close parentheses close square brackets over denominator open parentheses x cubed minus 2 close parentheses to the power of 4 end fraction equals

      equals fraction numerator open parentheses 12 x to the power of 5 minus 48 x squared close parentheses times open parentheses x cubed minus 2 close parentheses squared space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space 6 x squared open parentheses x cubed minus 2 close parentheses over denominator open parentheses x cubed minus 2 close parentheses to the power of 4 end fraction equals

      equals fraction numerator open parentheses x cubed minus 2 close parentheses open square brackets open parentheses 12 x to the power of 5 minus 48 x squared close parentheses times open parentheses x cubed minus 2 close parentheses space minus space open parentheses 2 x to the power of 6 minus 16 x cubed close parentheses times space 6 x squared close square brackets over denominator open parentheses x cubed minus 2 close parentheses to the power of 4 end fraction equals

      equals fraction numerator 12 x to the power of 8 minus 24 x to the power of 5 minus 48 x to the power of 5 plus 96 x squared minus space 12 x to the power of 8 plus 96 x to the power of 5 over denominator open parentheses x cubed minus 2 close parentheses cubed end fraction equals fraction numerator 24 x to the power of 5 plus 96 x squared over denominator open parentheses x cubed minus 2 close parentheses cubed end fraction equals fraction numerator bold 24 bold x to the power of bold 2 open parentheses bold x to the power of bold 3 bold plus bold 4 close parentheses over denominator open parentheses bold x to the power of bold 3 bold minus bold 2 close parentheses to the power of bold 3 end fraction

  8. Ania pisze:

    Cześć!Mam problem z pochodną 6x(x^2+1)^2 mógłbys wytłumaczyc krok po kroku?

  9. Ewusia pisze:

    Witam serdecznie. Mam problem z pochodną f(x)= 3/((1-x^2)(1-2x^3)). Kalkulator pokazuje odpowiedź: 6x(-5x^3+3x+1)/(mianownik^2).  A w moich obliczeniach wszystko się zgadza oprócz tego, że mam -6x. Ktoś wie co się stało z tym minusem? Proszę o odpowiedź

  10. Anna pisze:

    Witam. Mam problem z policzeniem pochodnej f(x)=ln(x)log_2(x)

  11. Agata pisze:

    Witam, nie rozumiem dlaczego pochodna z funkcji f(x)=e^2x+e^-x wychodzi e^2-e^-x a nie 2e^2x-e^-xBardzo proszę o odp 

  12. Robert pisze:

    Witammam policzyć  pochodne i nie potrafię sobie z nimi poradzić:mogę prosić o pomoc   

  13. Lidia pisze:

    dzień dobry,czy ktoś może wie w jaki sposób krok po kroku obliczyć pochodną poniższej funkcji?y equals fraction numerator x cubed sin open parentheses fourth root of 3 x end root close parentheses over denominator cos open parentheses x close parentheses end fractionBędę wdzięczna za pomoc :)0e/4f/f953a01e1eb0d9930b2c7913acbd.png” alt=”cos left parenthesis 2 x right parenthesis equals cos squared x minus sin squared x space equals space 2 cos squared x minus 1 space equals space 1 minus 2 sin squared x” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mo»(«/mo»«mn»2«/mn»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mo»-«/mo»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»2«/mn»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mo»-«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»1«/mn»«mo»-«/mo»«mn»2«/mn»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«/math»” /> – wykorzystana została wersja pierwsza.

    Rozpisując Pani wynik: 

    8 sin squared x space – space 8 cos squared x equals negative 8 times open parentheses negative sin squared x space plus cos squared x close parentheses equals negative 8 open parentheses bold italic c bold italic o bold italic s to the power of bold 2 bold italic x bold minus bold italic s bold italic i bold italic n to the power of bold 2 bold italic x close parentheses equals negative 8 bold italic c bold italic o bold italic s bold left parenthesis bold 2 bold italic x bold right parenthesis

    fa/71/2b12339d69cafcccfc9e75245a27.png” alt=”5 over 3 times x to the power of 2 over 3 end exponent times ln x plus x to the power of begin display style 5 over 3 end style end exponent over x equals 5 over 3 times x to the power of 2 over 3 end exponent times ln x plus x to the power of 5 over 3 minus 1 end exponent equals 5 over 3 times x to the power of 2 over 3 end exponent times ln x plus x to the power of 2 over 3 end exponent equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»5«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mi»x«/mi»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»+«/mo»«mfrac»«msup»«mi»x«/mi»«mstyle displaystyle=¨true¨»«mfrac»«mn»5«/mn»«mn»3«/mn»«/mfrac»«/mstyle»«/msup»«mi»x«/mi»«/mfrac»«mo»=«/mo»«mfrac»«mn»5«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mi»x«/mi»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»+«/mo»«msup»«mi»x«/mi»«mrow»«mfrac»«mn»5«/mn»«mn»3«/mn»«/mfrac»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«mo»=«/mo»«mfrac»«mn»5«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mi»x«/mi»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»+«/mo»«msup»«mi»x«/mi»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/msup»«mo»=«/mo»«/math»” />

    x to the power of 2 over 3 end exponent times open parentheses 5 over 3 ln x plus 1 close parentheses equals cube root of x squared end root times open parentheses 5 over 3 ln x plus 1 close parentheses

    ce/50/e97f118f5da367c319b5e5b5e657.png” alt=”3 times open parentheses e to the power of 2 x end exponent times 2 times ln x plus e to the power of 2 x end exponent times 1 over x close parentheses equals 3 times e to the power of 2 x end exponent times open parentheses 2 ln x plus 1 over x close parentheses” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mo»§#183;«/mo»«mfenced»«mrow»«msup»«mi»e«/mi»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«/msup»«mo»§#183;«/mo»«mn»2«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»+«/mo»«msup»«mi»e«/mi»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«mo»=«/mo»«mn»3«/mn»«mo»§#183;«/mo»«msup»«mi»e«/mi»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«/msup»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi»ln«/mi»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«/math»” />c5/4c/1333694f249760b916894e02b963.png” alt=”times fraction numerator 2 cos x times open parentheses cos x close parentheses apostrophe times open parentheses x cubed minus 3 x close parentheses minus cos squared x times open parentheses 3 x squared minus 3 close parentheses over denominator open parentheses x cubed minus 3 x close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#183;«/mo»«mfrac»«mrow»«mn»2«/mn»«mi»cos«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»`«/mo»«mo»§#183;«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mo»-«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    1 fifth times fifth root of open parentheses fraction numerator x cubed minus 3 x over denominator cos squared x end fraction close parentheses to the power of 4 end root times fraction numerator 2 cos x times open parentheses negative sin x close parentheses times open parentheses x cubed minus 3 x close parentheses minus cos squared x times open parentheses 3 x squared minus 3 close parentheses over denominator open parentheses x cubed minus 3 x close parentheses squared end fraction equals

    1 fifth times fraction numerator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 4 end root over denominator fifth root of open parentheses cos squared x close parentheses to the power of 4 end root end fraction times fraction numerator negative cos x times open square brackets 2 sin x times open parentheses x cubed minus 3 x close parentheses plus cos x times open parentheses 3 x squared minus 3 close parentheses close square brackets over denominator fifth root of open parentheses open parentheses x cubed minus 3 x close parentheses squared close parentheses to the power of 5 end root end fraction equals

    equals negative 1 fifth times fraction numerator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 4 end root over denominator fifth root of cos to the power of 8 x end root end fraction times fraction numerator fifth root of cos to the power of 5 x end root times open square brackets 2 times open parentheses x cubed minus 3 x close parentheses times sin x plus open parentheses 3 x squared minus 3 close parentheses times cos x close square brackets over denominator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 10 end root end fraction equals

    equals negative 1 fifth times fraction numerator 2 times open parentheses x cubed minus 3 x close parentheses times sin x plus open parentheses 3 x squared minus 3 close parentheses times cos x over denominator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 6 times cos cubed x end root end fraction

    dc/9d/43f65dd64634765de9142c72d807.png” alt=”4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«mfrac»«mrow»«mo»-«/mo»«mn»25«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»-«/mo»«mn»10«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«/mrow»«mrow»«msup»«mi»x«/mi»«mstyle displaystyle=¨true¨»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mstyle»«/msup»«mo»§#183;«/mo»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/math»” />

     

    02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

    x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

    x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

    capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
    x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
    x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
    wykres

    Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
    Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

     

    Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

     

    Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  14. Kasiek pisze:

    Dzień dobry,zasanowiła mnie jedna rzecz. Chcąc sprawdzić wynik pochodnej (-8cos(x)sin(x))’ znalałzam Pana kalkulator i inny. wg Pana kalkulatora wynik to (-8cos(2x)), a to wyszło w innym  (8(sinx)^2 – 8(cosx)^2) – i ja też otrzymałam taki wynik. Mogę prosić o pomoc?To całe zadanie jaki muszę obliczyć: -8cos(x)sin(x)+(e^(x^(1/2))(1- (1/x^(1/2))) /(4x))”Podzieliłam” je na 2 zgodnie z właściwościami pochodnych – [f(x)+g(x)]’ = f'(x)+g'(x) 

    1. “wg Pana kalkulatora wynik to (-8cos(2x)), a to wyszło w innym  (8(sinx)^2 – 8(cosx)^2) – i ja też otrzymałam taki wynik.”

      Pani Kasiu – oba wyniki są poprawne 🙂 Policzyła Pani wszystko prawidłowo.

      Kalkulator zamieszczony na Blogu po prostu dodatkowo dokonał jeszcze jedne przekształcenie, wykorzystując rozpisanie wzoru cos left parenthesis 2 x right parenthesis ze szkoły średniej (jak pamiętamy, tam były jego 3 wersje)

      cos left parenthesis 2 x right parenthesis equals cos squared x minus sin squared x space equals space 2 cos squared x minus 1 space equals space 1 minus 2 sin squared x – wykorzystana została wersja pierwsza.

      Rozpisując Pani wynik: 

      8 sin squared x space – space 8 cos squared x equals negative 8 times open parentheses negative sin squared x space plus cos squared x close parentheses equals negative 8 open parentheses bold italic c bold italic o bold italic s to the power of bold 2 bold italic x bold minus bold italic s bold italic i bold italic n to the power of bold 2 bold italic x close parentheses equals negative 8 bold italic c bold italic o bold italic s bold left parenthesis bold 2 bold italic x bold right parenthesis

      fa/71/2b12339d69cafcccfc9e75245a27.png” alt=”5 over 3 times x to the power of 2 over 3 end exponent times ln x plus x to the power of begin display style 5 over 3 end style end exponent over x equals 5 over 3 times x to the power of 2 over 3 end exponent times ln x plus x to the power of 5 over 3 minus 1 end exponent equals 5 over 3 times x to the power of 2 over 3 end exponent times ln x plus x to the power of 2 over 3 end exponent equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»5«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mi»x«/mi»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»+«/mo»«mfrac»«msup»«mi»x«/mi»«mstyle displaystyle=¨true¨»«mfrac»«mn»5«/mn»«mn»3«/mn»«/mfrac»«/mstyle»«/msup»«mi»x«/mi»«/mfrac»«mo»=«/mo»«mfrac»«mn»5«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mi»x«/mi»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»+«/mo»«msup»«mi»x«/mi»«mrow»«mfrac»«mn»5«/mn»«mn»3«/mn»«/mfrac»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«mo»=«/mo»«mfrac»«mn»5«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mi»x«/mi»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»+«/mo»«msup»«mi»x«/mi»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/msup»«mo»=«/mo»«/math»” />

      x to the power of 2 over 3 end exponent times open parentheses 5 over 3 ln x plus 1 close parentheses equals cube root of x squared end root times open parentheses 5 over 3 ln x plus 1 close parentheses

      ce/50/e97f118f5da367c319b5e5b5e657.png” alt=”3 times open parentheses e to the power of 2 x end exponent times 2 times ln x plus e to the power of 2 x end exponent times 1 over x close parentheses equals 3 times e to the power of 2 x end exponent times open parentheses 2 ln x plus 1 over x close parentheses” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mo»§#183;«/mo»«mfenced»«mrow»«msup»«mi»e«/mi»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«/msup»«mo»§#183;«/mo»«mn»2«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»+«/mo»«msup»«mi»e«/mi»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«mo»=«/mo»«mn»3«/mn»«mo»§#183;«/mo»«msup»«mi»e«/mi»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«/msup»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi»ln«/mi»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«/math»” />c5/4c/1333694f249760b916894e02b963.png” alt=”times fraction numerator 2 cos x times open parentheses cos x close parentheses apostrophe times open parentheses x cubed minus 3 x close parentheses minus cos squared x times open parentheses 3 x squared minus 3 close parentheses over denominator open parentheses x cubed minus 3 x close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#183;«/mo»«mfrac»«mrow»«mn»2«/mn»«mi»cos«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»`«/mo»«mo»§#183;«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mo»-«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      1 fifth times fifth root of open parentheses fraction numerator x cubed minus 3 x over denominator cos squared x end fraction close parentheses to the power of 4 end root times fraction numerator 2 cos x times open parentheses negative sin x close parentheses times open parentheses x cubed minus 3 x close parentheses minus cos squared x times open parentheses 3 x squared minus 3 close parentheses over denominator open parentheses x cubed minus 3 x close parentheses squared end fraction equals

      1 fifth times fraction numerator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 4 end root over denominator fifth root of open parentheses cos squared x close parentheses to the power of 4 end root end fraction times fraction numerator negative cos x times open square brackets 2 sin x times open parentheses x cubed minus 3 x close parentheses plus cos x times open parentheses 3 x squared minus 3 close parentheses close square brackets over denominator fifth root of open parentheses open parentheses x cubed minus 3 x close parentheses squared close parentheses to the power of 5 end root end fraction equals

      equals negative 1 fifth times fraction numerator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 4 end root over denominator fifth root of cos to the power of 8 x end root end fraction times fraction numerator fifth root of cos to the power of 5 x end root times open square brackets 2 times open parentheses x cubed minus 3 x close parentheses times sin x plus open parentheses 3 x squared minus 3 close parentheses times cos x close square brackets over denominator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 10 end root end fraction equals

      equals negative 1 fifth times fraction numerator 2 times open parentheses x cubed minus 3 x close parentheses times sin x plus open parentheses 3 x squared minus 3 close parentheses times cos x over denominator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 6 times cos cubed x end root end fraction

      dc/9d/43f65dd64634765de9142c72d807.png” alt=”4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«mfrac»«mrow»«mo»-«/mo»«mn»25«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»-«/mo»«mn»10«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«/mrow»«mrow»«msup»«mi»x«/mi»«mstyle displaystyle=¨true¨»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mstyle»«/msup»«mo»§#183;«/mo»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/math»” />

       

      02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  15. Bubi pisze:

    mam problem z pochodną funkcji : left parenthesis 1 plus square root of x right parenthesis to the power of ln square root of x end exponent56/84/95809cc411264007185aa38dc9ae.png” alt=”e to the power of c o ś end exponent” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»e«/mi»«mrow»«mi»c«/mi»«mi»o«/mi»«mi»§#347;«/mi»«/mrow»«/msup»«/math»” />. 
    Postępujemy jak zawsze w takich przypadkach, czyli: pochodna tego co „na zewnątrz” pomnożyć razy pochodna funkcji wewnętrznej (coś więcej niż sam „x”), czyli jakby open parentheses e to the power of increment close parentheses apostrophe equals e to the power of increment times increment apostrophe .

    Stąd: open parentheses e to the power of 8 x end exponent close parentheses apostrophe equals e to the power of 8 x end exponent times open parentheses 8 x close parentheses apostrophe equals e to the power of 8 x end exponent times 8 times 1 equals 8 e to the power of 8 x end exponent 

    16/ff/8470c5a592a295d322b9588044e0.png” alt=”open parentheses a to the power of x close parentheses apostrophe equals a to the power of x times ln a” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«msup»«mi»a«/mi»«mi»x«/mi»«/msup»«/mfenced»«mo»`«/mo»«mo»=«/mo»«msup»«mi»a«/mi»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mi»a«/mi»«/math»” /> .56/59/da565d75ad307a420ee679d5b107.png” alt=”a r c sin x plus a r c cos x equals C equals a r c sin 0 plus a r c cos 0 equals 0 plus pi over 2 equals pi over 2″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»=«/mo»«mi»C«/mi»«mo»=«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mn»0«/mn»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mn»0«/mn»«mo»=«/mo»«mn»0«/mn»«mo»+«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«/math»” />

    Wtedy funkcja

    y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r c sin x plus a r c cos x close vertical bar end root equals fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 5,

    i jej pochodna

    (wg wzoru dla funkcji złożonej:  open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 triangle to the power of 4 times open parentheses triangle close parentheses apostrophe   )

    wynosi:

    y apostrophe equals fifth root of pi over 2 end root times 5 times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses sin x plus cos x close parentheses apostrophe equals

    fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses

     

    b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

     

    02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

    x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

    x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

    capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
    x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
    x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
    wykres

    Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
    Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

     

    Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

     

    Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  16. kati pisze:

    square root of 1 minus 3 x hat 2 end root equals… ; 2 to the power of 3 x plus 4 end exponent ln x equals
… proszę o pomoc0a/73/31685f469bf7afc9fe6784206245.png” alt=”open parentheses ln left parenthesis increment right parenthesis close parentheses apostrophe equals 1 over increment times increment apostrophe” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mi»ln«/mi»«mo»(«/mo»«mo»§#8710;«/mo»«mo»)«/mo»«/mrow»«/mfenced»«mo»`«/mo»«mo»=«/mo»«mfrac»«mn»1«/mn»«mo»§#8710;«/mo»«/mfrac»«mo»§#183;«/mo»«mo»§#8710;«/mo»«mo»`«/mo»«/math»” />  , gdzie za ten increment biorę funkcję wewnętrzną.

    Stąd ostatecznie: open parentheses ln left parenthesis 2 x right parenthesis close parentheses apostrophe equals fraction numerator 1 over denominator 2 x end fraction times open parentheses 2 x close parentheses apostrophe equals fraction numerator 1 over denominator 2 x end fraction times 2 times 1 equals fraction numerator 2 over denominator 2 x end fraction equals 1 over x

    56/59/da565d75ad307a420ee679d5b107.png” alt=”a r c sin x plus a r c cos x equals C equals a r c sin 0 plus a r c cos 0 equals 0 plus pi over 2 equals pi over 2″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»=«/mo»«mi»C«/mi»«mo»=«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mn»0«/mn»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mn»0«/mn»«mo»=«/mo»«mn»0«/mn»«mo»+«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«/math»” />

    Wtedy funkcja

    y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r c sin x plus a r c cos x close vertical bar end root equals fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 5,

    i jej pochodna

    (wg wzoru dla funkcji złożonej:  open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 triangle to the power of 4 times open parentheses triangle close parentheses apostrophe   )

    wynosi:

    y apostrophe equals fifth root of pi over 2 end root times 5 times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses sin x plus cos x close parentheses apostrophe equals

    fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses

     

    b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

     

    02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

    x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

    x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

    capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
    x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
    x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
    wykres

    Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
    Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

     

    Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

     

    Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

    1. kati pisze:

      oczywiście polecenie policz pochodne 

    2. 1. y equals square root of 1 minus 3 x squared end root

      Stosuję wzór open parentheses square root of triangle close parentheses apostrophe equals fraction numerator 1 over denominator 2 square root of triangle end fraction times triangle apostrophe

      y apostrophe equals fraction numerator 1 over denominator 2 square root of 1 minus 3 x squared end root end fraction times open parentheses 1 minus 3 x squared close parentheses apostrophe equals fraction numerator 1 over denominator 2 square root of 1 minus 3 x squared end root end fraction times open parentheses 0 minus 3 times 2 x close parentheses equals

      fraction numerator negative 6 x over denominator 2 square root of 1 minus 3 x squared end root end fraction equals negative fraction numerator 3 x over denominator square root of 1 minus 3 x squared end root end fraction

      2. y equals 2 to the power of 3 x plus 4 end exponent times ln x

      Stosuję wzory: open parentheses u times v close parentheses apostrophe equals u apostrophe times v plus u times v apostrophe oraz open parentheses 2 to the power of triangle close parentheses apostrophe equals 2 to the power of triangle times ln 2 times triangle apostrophe

      y apostrophe equals open parentheses 2 to the power of 3 x plus 4 end exponent close parentheses apostrophe times ln x plus 2 to the power of 3 x plus 4 end exponent times open parentheses ln x close parentheses apostrophe equals 2 to the power of 3 x plus 4 end exponent times ln 2 times open parentheses 3 x plus 4 close parentheses apostrophe times ln x plus 2 to the power of 3 x plus 4 end exponent times 1 over x equals

      2 to the power of 3 x plus 4 end exponent times ln 2 times open parentheses 3 plus 0 close parentheses times ln x plus 2 to the power of 3 x plus 4 end exponent times 1 over x equals 2 to the power of 3 x plus 4 end exponent times open parentheses 3 ln 2 times ln x plus 1 over x close parentheses

      da/95/2ae01e3253ad8621bf6d29b2b3ed.png” alt=”fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mroot»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«mn»5«/mn»«/mroot»«mo»§#183;«/mo»«msup»«mfenced»«mrow»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mn»4«/mn»«/msup»«mo»§#183;«/mo»«mfenced»«mrow»«mi»cos«/mi»«mi»x«/mi»«mo»-«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mfenced»«/math»” />

       

      b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

       

      02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  17. kati pisze:

    space p r o s z ę space o space r a t u n e k space m a m space d o space p o l i c z e n i a space d w i e space p o c h o d n e space square root of 1 minus 3 x hat 2 end root space space2 to the power of 3 x plus 4 end exponent ln x0a/73/31685f469bf7afc9fe6784206245.png” alt=”open parentheses ln left parenthesis increment right parenthesis close parentheses apostrophe equals 1 over increment times increment apostrophe” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mi»ln«/mi»«mo»(«/mo»«mo»§#8710;«/mo»«mo»)«/mo»«/mrow»«/mfenced»«mo»`«/mo»«mo»=«/mo»«mfrac»«mn»1«/mn»«mo»§#8710;«/mo»«/mfrac»«mo»§#183;«/mo»«mo»§#8710;«/mo»«mo»`«/mo»«/math»” />  , gdzie za ten increment biorę funkcję wewnętrzną.

    Stąd ostatecznie: open parentheses ln left parenthesis 2 x right parenthesis close parentheses apostrophe equals fraction numerator 1 over denominator 2 x end fraction times open parentheses 2 x close parentheses apostrophe equals fraction numerator 1 over denominator 2 x end fraction times 2 times 1 equals fraction numerator 2 over denominator 2 x end fraction equals 1 over x

    56/59/da565d75ad307a420ee679d5b107.png” alt=”a r c sin x plus a r c cos x equals C equals a r c sin 0 plus a r c cos 0 equals 0 plus pi over 2 equals pi over 2″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»=«/mo»«mi»C«/mi»«mo»=«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mn»0«/mn»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mn»0«/mn»«mo»=«/mo»«mn»0«/mn»«mo»+«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«/math»” />

    Wtedy funkcja

    y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r c sin x plus a r c cos x close vertical bar end root equals fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 5,

    i jej pochodna

    (wg wzoru dla funkcji złożonej:  open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 triangle to the power of 4 times open parentheses triangle close parentheses apostrophe   )

    wynosi:

    y apostrophe equals fifth root of pi over 2 end root times 5 times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses sin x plus cos x close parentheses apostrophe equals

    fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses

     

    b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

     

    02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

    x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

    x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

    capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
    x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
    x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
    wykres

    Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
    Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

     

    Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

     

    Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  18. Kamil pisze:

    3 ln hat 5 left parenthesis 3 over x to the power of 4 minus x right parenthesisWitam mam problem z obliczeniem tej pochodnej mógłby mi ktoś wytłumaczyć jak to zrobić ?5c/e0/f62382dc5955d0be0c846a56ae3f.png” alt=”open parentheses square root of x squared end root close parentheses apostrophe equals open parentheses fraction numerator 1 over denominator 2 square root of x squared end root end fraction close parentheses times open parentheses x squared close parentheses apostrophe equals fraction numerator 2 x over denominator 2 square root of x squared end root end fraction equals fraction numerator x over denominator square root of x squared end root end fraction equals fraction numerator x over denominator open vertical bar x close vertical bar end fraction equals open curly brackets table attributes columnalign left end attributes row cell 1 space space space space space space d l a space x greater or equal than 0 end cell row cell negative 1 space space space d l a space x less than 0 end cell end table close” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfenced»«mo»`«/mo»«mo»=«/mo»«mfenced»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mrow»«/mfrac»«/mfenced»«mo»§#183;«/mo»«mfenced»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfenced»«mo»`«/mo»«mo»=«/mo»«mfrac»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«mrow»«mn»2«/mn»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mi»x«/mi»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mo»=«/mo»«mfrac»«mi»x«/mi»«mfenced open=¨|¨ close=¨|¨»«mi»x«/mi»«/mfenced»«/mfrac»«mo»=«/mo»«mfenced open=¨{¨ close=¨¨»«mtable columnalign=¨left¨»«mtr»«mtd»«mn»1«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»d«/mi»«mi»l«/mi»«mi»a«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo»-«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»d«/mi»«mi»l«/mi»«mi»a«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mfenced»«/math»” />0a/73/31685f469bf7afc9fe6784206245.png” alt=”open parentheses ln left parenthesis increment right parenthesis close parentheses apostrophe equals 1 over increment times increment apostrophe” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mi»ln«/mi»«mo»(«/mo»«mo»§#8710;«/mo»«mo»)«/mo»«/mrow»«/mfenced»«mo»`«/mo»«mo»=«/mo»«mfrac»«mn»1«/mn»«mo»§#8710;«/mo»«/mfrac»«mo»§#183;«/mo»«mo»§#8710;«/mo»«mo»`«/mo»«/math»” />  , gdzie za ten increment biorę funkcję wewnętrzną.

    Stąd ostatecznie: open parentheses ln left parenthesis 2 x right parenthesis close parentheses apostrophe equals fraction numerator 1 over denominator 2 x end fraction times open parentheses 2 x close parentheses apostrophe equals fraction numerator 1 over denominator 2 x end fraction times 2 times 1 equals fraction numerator 2 over denominator 2 x end fraction equals 1 over x

    56/59/da565d75ad307a420ee679d5b107.png” alt=”a r c sin x plus a r c cos x equals C equals a r c sin 0 plus a r c cos 0 equals 0 plus pi over 2 equals pi over 2″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»=«/mo»«mi»C«/mi»«mo»=«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mn»0«/mn»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mn»0«/mn»«mo»=«/mo»«mn»0«/mn»«mo»+«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«/math»” />

    Wtedy funkcja

    y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r c sin x plus a r c cos x close vertical bar end root equals fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 5,

    i jej pochodna

    (wg wzoru dla funkcji złożonej:  open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 triangle to the power of 4 times open parentheses triangle close parentheses apostrophe   )

    wynosi:

    y apostrophe equals fifth root of pi over 2 end root times 5 times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses sin x plus cos x close parentheses apostrophe equals

    fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses

     

    b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

     

    02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

    x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

    x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

    capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
    x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
    x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
    wykres

    Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
    Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

     

    Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

     

    Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

    1. y equals 3 ln to the power of 5 open parentheses 3 over x to the power of 4 minus x close parentheses

      Stosuję wzory open parentheses C times f open parentheses x close parentheses close parentheses apostrophe equals C times f apostrophe open parentheses x close parentheses oraz open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 times triangle to the power of 4 times triangle apostrophe

      y apostrophe equals 3 times open square brackets ln to the power of 5 open parentheses 3 over x to the power of 4 minus x close parentheses close square brackets apostrophe equals 3 times 5 times ln to the power of 4 open parentheses 3 over x to the power of 4 minus x close parentheses times open parentheses 3 over x to the power of 4 minus x close parentheses apostrophe equals

      15 ln to the power of 4 open parentheses 3 over x to the power of 4 minus x close parentheses times open parentheses 3 times x to the power of negative 4 end exponent minus x close parentheses apostrophe equals 15 ln to the power of 4 open parentheses 3 over x to the power of 4 minus x close parentheses times open parentheses 3 times open parentheses negative 4 close parentheses times x to the power of negative 5 end exponent minus 1 close parentheses equals

      15 ln to the power of 4 open parentheses 3 over x to the power of 4 minus x close parentheses times open parentheses negative 12 over x to the power of 5 minus 1 close parentheses equals negative 15 times open parentheses 12 over x to the power of 5 plus 1 close parentheses times ln to the power of 4 open parentheses 3 over x to the power of 4 minus x close parentheses

      2b/44/2d219e11a8441bbf9ac9e4e1d087.png” alt=”y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r c sin x plus a r c cos x close vertical bar end root equals fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 5″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«msup»«mfenced»«mrow»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mn»5«/mn»«/msup»«mo»§#183;«/mo»«mroot»«mfenced open=¨|¨ close=¨|¨»«mrow»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mn»5«/mn»«/mroot»«mo»=«/mo»«mroot»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«mn»5«/mn»«/mroot»«mo»§#183;«/mo»«msup»«mfenced»«mrow»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mn»5«/mn»«/msup»«/math»” />,

      i jej pochodna

      (wg wzoru dla funkcji złożonej:  open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 triangle to the power of 4 times open parentheses triangle close parentheses apostrophe   )

      wynosi:

      y apostrophe equals fifth root of pi over 2 end root times 5 times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses sin x plus cos x close parentheses apostrophe equals

      fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses

       

      b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

       

      02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  19. Mati pisze:

    w kalkulatorze wychodzą bzdury gdy liczy się pochodną pierwiastków:np po wpisaniu (x^2)^-2 czyli square root of cross times squared end root (pochodna to oczywiście 1) wychodzi75/87/9632652090901cb4f3517a326e72.png” alt=”left parenthesis f left parenthesis g left parenthesis x right parenthesis right parenthesis apostrophe space equals space f apostrophe left parenthesis g left parenthesis x right parenthesis right parenthesis space times space g apostrophe left parenthesis x right parenthesis” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«mi»f«/mi»«mo»(«/mo»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»)«/mo»«mo»`«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»)«/mo»«mo»§#160;«/mo»«mo»§#183;«/mo»«mo»§#160;«/mo»«mi»g«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»” />

    Przy naszych danych to pójdzie tak: open parentheses ln left parenthesis increment right parenthesis close parentheses apostrophe equals 1 over increment times increment apostrophe  , gdzie za ten increment biorę funkcję wewnętrzną.

    Stąd ostatecznie: open parentheses ln left parenthesis 2 x right parenthesis close parentheses apostrophe equals fraction numerator 1 over denominator 2 x end fraction times open parentheses 2 x close parentheses apostrophe equals fraction numerator 1 over denominator 2 x end fraction times 2 times 1 equals fraction numerator 2 over denominator 2 x end fraction equals 1 over x

    56/59/da565d75ad307a420ee679d5b107.png” alt=”a r c sin x plus a r c cos x equals C equals a r c sin 0 plus a r c cos 0 equals 0 plus pi over 2 equals pi over 2″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»=«/mo»«mi»C«/mi»«mo»=«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mn»0«/mn»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mn»0«/mn»«mo»=«/mo»«mn»0«/mn»«mo»+«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«/math»” />

    Wtedy funkcja

    y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r c sin x plus a r c cos x close vertical bar end root equals fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 5,

    i jej pochodna

    (wg wzoru dla funkcji złożonej:  open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 triangle to the power of 4 times open parentheses triangle close parentheses apostrophe   )

    wynosi:

    y apostrophe equals fifth root of pi over 2 end root times 5 times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses sin x plus cos x close parentheses apostrophe equals

    fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses

     

    b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

     

    02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

    x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

    x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

    capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
    x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
    x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
    wykres

    Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
    Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

     

    Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

     

    Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

    1. Tutaj akurat kalkulator dobrze policzył pochodną 🙂

      Wpisana formuła “(x^2)^-2” (potęga (-2) ) nie oznacza pierwiastka, tylko inna potęgę, a mianowicie:
      open parentheses x squared close parentheses to the power of negative 2 end exponent equals open parentheses 1 over x squared close parentheses squared equals 1 over x to the power of 4 equals x to the power of negative 4 end exponent – minus w potędze odwraca podstawę 🙂

      Aby wprowadzić pierwiastek, trzeba wziąć potęgę ułamkową, czyli powinien Pan wpisać “”(x^2)^(1/2)” 

      Wtedy pochodna:

      open parentheses square root of x squared end root close parentheses apostrophe equals open parentheses fraction numerator 1 over denominator 2 square root of x squared end root end fraction close parentheses times open parentheses x squared close parentheses apostrophe equals fraction numerator 2 x over denominator 2 square root of x squared end root end fraction equals fraction numerator x over denominator square root of x squared end root end fraction equals fraction numerator x over denominator open vertical bar x close vertical bar end fraction equals open curly brackets table attributes columnalign left end attributes row cell 1 space space space space space space d l a space x greater or equal than 0 end cell row cell negative 1 space space space d l a space x less than 0 end cell end table close

      0a/73/31685f469bf7afc9fe6784206245.png” alt=”open parentheses ln left parenthesis increment right parenthesis close parentheses apostrophe equals 1 over increment times increment apostrophe” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mi»ln«/mi»«mo»(«/mo»«mo»§#8710;«/mo»«mo»)«/mo»«/mrow»«/mfenced»«mo»`«/mo»«mo»=«/mo»«mfrac»«mn»1«/mn»«mo»§#8710;«/mo»«/mfrac»«mo»§#183;«/mo»«mo»§#8710;«/mo»«mo»`«/mo»«/math»” />  , gdzie za ten increment biorę funkcję wewnętrzną.

      Stąd ostatecznie: open parentheses ln left parenthesis 2 x right parenthesis close parentheses apostrophe equals fraction numerator 1 over denominator 2 x end fraction times open parentheses 2 x close parentheses apostrophe equals fraction numerator 1 over denominator 2 x end fraction times 2 times 1 equals fraction numerator 2 over denominator 2 x end fraction equals 1 over x

      56/59/da565d75ad307a420ee679d5b107.png” alt=”a r c sin x plus a r c cos x equals C equals a r c sin 0 plus a r c cos 0 equals 0 plus pi over 2 equals pi over 2″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»=«/mo»«mi»C«/mi»«mo»=«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mn»0«/mn»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mn»0«/mn»«mo»=«/mo»«mn»0«/mn»«mo»+«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«/math»” />

      Wtedy funkcja

      y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r c sin x plus a r c cos x close vertical bar end root equals fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 5,

      i jej pochodna

      (wg wzoru dla funkcji złożonej:  open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 triangle to the power of 4 times open parentheses triangle close parentheses apostrophe   )

      wynosi:

      y apostrophe equals fifth root of pi over 2 end root times 5 times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses sin x plus cos x close parentheses apostrophe equals

      fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses

       

      b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

       

      02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  20. Paulina pisze:

    Witam Panie Krzysztofie,czy mógłby mi Pan pomóc z obliczeniem pochodnej:f left parenthesis x right parenthesis equals fifth root of fraction numerator cos squared x over denominator x cubed minus 3 x end fraction end root1e/ed/d6c84666015ead1b00f59ae14c5a.png” alt=”C apostrophe equals 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«mo»`«/mo»«mo»=«/mo»«mn»0«/mn»«/math»” />, o ile C equals c o n s t space open parentheses s t a ł a close parentheses

    Obliczymy:

    open parentheses a r c sin x plus a r c cos x close parentheses apostrophe equals open parentheses a r c sin x close parentheses apostrophe plus open parentheses a r c cos x close parentheses apostrophe equals fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction plus open parentheses negative fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction close parentheses equals 0

    Stąd mamy, że a r c sin x plus a r c cos x equals C

    Liczba stała nie zależy od x. Obliczymy ją:

    a r c sin x plus a r c cos x equals C equals a r c sin 0 plus a r c cos 0 equals 0 plus pi over 2 equals pi over 2

    Wtedy funkcja

    y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r c sin x plus a r c cos x close vertical bar end root equals fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 5,

    i jej pochodna

    (wg wzoru dla funkcji złożonej:  open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 triangle to the power of 4 times open parentheses triangle close parentheses apostrophe   )

    wynosi:

    y apostrophe equals fifth root of pi over 2 end root times 5 times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses sin x plus cos x close parentheses apostrophe equals

    fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses

     

    b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

     

    02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

    x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

    x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

    capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
    x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
    x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
    wykres

    Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
    Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

     

    Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

     

    Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

    1. y equals fifth root of fraction numerator cos squared x over denominator x cubed minus 3 x end fraction end root

      Stosuję wzory dla pochodnej ułamku:

      open parentheses u over v close parentheses apostrophe equals fraction numerator u apostrophe times v minus u times v apostrophe over denominator v squared end fraction oraz pochodnej funkcji złożonej: 

      open square brackets f open parentheses g open parentheses x close parentheses close parentheses close square brackets apostrophe equals f apostrophe open parentheses g close parentheses times g apostrophe open parentheses x close parentheses

      y equals fifth root of fraction numerator cos squared x over denominator x cubed minus 3 x end fraction end root equals open parentheses fraction numerator cos squared x over denominator x cubed minus 3 x end fraction close parentheses to the power of 1 fifth end exponent. Wtedy

      y apostrophe equals 1 fifth times open parentheses fraction numerator cos squared x over denominator x cubed minus 3 x end fraction close parentheses to the power of 1 fifth minus 1 end exponent times open parentheses fraction numerator cos squared x over denominator x cubed minus 3 x end fraction close parentheses apostrophe equals 1 fifth times open parentheses fraction numerator cos squared x over denominator x cubed minus 3 x end fraction close parentheses to the power of negative 4 over 5 end exponent times

      times fraction numerator open parentheses cos squared x close parentheses apostrophe times open parentheses x cubed minus 3 x close parentheses minus cos squared x times open parentheses x cubed minus 3 x close parentheses apostrophe over denominator open parentheses x cubed minus 3 x close parentheses squared end fraction equals 1 fifth times fraction numerator 1 over denominator fifth root of open parentheses begin display style fraction numerator cos squared x over denominator x cubed minus 3 x end fraction end style close parentheses to the power of 4 end root end fraction times

      times fraction numerator 2 cos x times open parentheses cos x close parentheses apostrophe times open parentheses x cubed minus 3 x close parentheses minus cos squared x times open parentheses 3 x squared minus 3 close parentheses over denominator open parentheses x cubed minus 3 x close parentheses squared end fraction equals

      1 fifth times fifth root of open parentheses fraction numerator x cubed minus 3 x over denominator cos squared x end fraction close parentheses to the power of 4 end root times fraction numerator 2 cos x times open parentheses negative sin x close parentheses times open parentheses x cubed minus 3 x close parentheses minus cos squared x times open parentheses 3 x squared minus 3 close parentheses over denominator open parentheses x cubed minus 3 x close parentheses squared end fraction equals

      1 fifth times fraction numerator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 4 end root over denominator fifth root of open parentheses cos squared x close parentheses to the power of 4 end root end fraction times fraction numerator negative cos x times open square brackets 2 sin x times open parentheses x cubed minus 3 x close parentheses plus cos x times open parentheses 3 x squared minus 3 close parentheses close square brackets over denominator fifth root of open parentheses open parentheses x cubed minus 3 x close parentheses squared close parentheses to the power of 5 end root end fraction equals

      equals negative 1 fifth times fraction numerator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 4 end root over denominator fifth root of cos to the power of 8 x end root end fraction times fraction numerator fifth root of cos to the power of 5 x end root times open square brackets 2 times open parentheses x cubed minus 3 x close parentheses times sin x plus open parentheses 3 x squared minus 3 close parentheses times cos x close square brackets over denominator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 10 end root end fraction equals

      equals negative 1 fifth times fraction numerator 2 times open parentheses x cubed minus 3 x close parentheses times sin x plus open parentheses 3 x squared minus 3 close parentheses times cos x over denominator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 6 times cos cubed x end root end fraction

      dc/9d/43f65dd64634765de9142c72d807.png” alt=”4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«mfrac»«mrow»«mo»-«/mo»«mn»25«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»-«/mo»«mn»10«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«/mrow»«mrow»«msup»«mi»x«/mi»«mstyle displaystyle=¨true¨»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mstyle»«/msup»«mo»§#183;«/mo»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/math»” />

       

      02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  21. Mike pisze:

    Dzień dobry, chciałem zwrócić uwagę na błąd, gdy w pochodnej funkcji sqrt(3^3 -2) wynikiem jest ((3^x)log(3))/(2(sqrt(3x-2))), gdzie w miejscu log powinno być ln.Pozdrawiam

  22. Justyna pisze:

    Witam, w ostatniej lekcji z kursu pochodnych robił Pan przykład x/lnx, Moje pytanie brzmi skąd w wykresie 2 pochodnej wziął się punkt 1. wklejam juz policzoną 2 pochodną

  23. Mateusz pisze:

    Dzień dobry Panie Krystianie!Czy mógłby mi Pan pomóc w obliczeniu pochodnej z: left parenthesis sin x plus cos x right parenthesis to the power of 5* fifth root of vertical line a r c sin x plus a r c cos x vertical line end root?3f/e2/bbc6138a3c0a4628acc40a28f1c7.png” alt=”capital delta equals left parenthesis negative 1 right parenthesis squared minus 4 times 1 times left parenthesis negative 2 right parenthesis equals 9″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#916;«/mi»«mo»=«/mo»«mo»(«/mo»«mo»-«/mo»«mn»1«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»4«/mn»«mo»§#183;«/mo»«mn»1«/mn»«mo»§#183;«/mo»«mo»(«/mo»«mo»-«/mo»«mn»2«/mn»«mo»)«/mo»«mo»=«/mo»«mn»9«/mn»«/math»”>
    x subscript 1 equals fraction numerator 1 minus square root of 9 over denominator 2 end fraction equals fraction numerator 1 minus 3 over denominator 2 end fraction equals fraction numerator negative 2 over denominator 2 end fraction equals negative 1
    space x subscript 2 equals fraction numerator 1 plus square root of 9 over denominator 2 end fraction equals fraction numerator 1 plus 3 over denominator 2 end fraction equals 4 over 2 equals 2

    Zatem D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

    Przechodzimy do wyznaczania monotoniczności funkcji f. W tym celu obliczymy jej pochodną i sprawdzimy, kiedy jest dodatnia, a kiedy ujemna.

    f apostrophe left parenthesis x right parenthesis equals fraction numerator open parentheses x cubed close parentheses apostrophe times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses x squared minus x minus 2 close parentheses apostrophe over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals
    equals fraction numerator 3 x squared times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses 2 x minus 1 close parentheses over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals fraction numerator 3 x to the power of 4 minus 3 x cubed minus 6 x squared minus 2 x to the power of 4 plus x cubed over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals
    equals fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction

    Zbadamy teraz, kiedy pochodna przyjmuje wartości większe lub równe 0, a kiedy mniejsze lub równe 0.

    fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0

    x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

    x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

    capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
    x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
    x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
    wykres

    Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
    Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

     

    Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

     

    Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

    1. y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r s c i n x plus a r c cos x close vertical bar end root

      Wiadomo, że pochodna liczby stałej wynosi zero:

      C apostrophe equals 0, o ile C equals c o n s t space open parentheses s t a ł a close parentheses

      Obliczymy:

      open parentheses a r c sin x plus a r c cos x close parentheses apostrophe equals open parentheses a r c sin x close parentheses apostrophe plus open parentheses a r c cos x close parentheses apostrophe equals fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction plus open parentheses negative fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction close parentheses equals 0

      Stąd mamy, że a r c sin x plus a r c cos x equals C

      Liczba stała nie zależy od x. Obliczymy ją:

      a r c sin x plus a r c cos x equals C equals a r c sin 0 plus a r c cos 0 equals 0 plus pi over 2 equals pi over 2

      Wtedy funkcja

      y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r c sin x plus a r c cos x close vertical bar end root equals fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 5,

      i jej pochodna

      (wg wzoru dla funkcji złożonej:  open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 triangle to the power of 4 times open parentheses triangle close parentheses apostrophe   )

      wynosi:

      y apostrophe equals fifth root of pi over 2 end root times 5 times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses sin x plus cos x close parentheses apostrophe equals

      fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses

       

      b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

       

      02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  24. Kasia pisze:

    Witam, potrzebuję obliczyć pierwszą pochodną funkcji. Jak to zrobić?i(x) = 3e^2x *lnx

    1. y equals 3 e to the power of 2 x end exponent times ln x

      Stosuję wzory: open parentheses C times y close parentheses apostrophe equals C times y apostrophe (gdzie C – stała) oraz open parentheses u times v close parentheses apostrophe equals u apostrophe times v plus u times v apostrophe, a także

      open parentheses e to the power of triangle close parentheses apostrophe equals e to the power of triangle times triangle apostrophe

      y apostrophe equals 3 times open square brackets open parentheses e to the power of 2 x end exponent close parentheses apostrophe times ln x plus e to the power of 2 x end exponent times open parentheses ln x close parentheses apostrophe close square brackets equals 3 times open square brackets e to the power of 2 x end exponent times open parentheses 2 x close parentheses apostrophe times ln x plus e to the power of 2 x end exponent times 1 over x close square brackets equals

      3 times open parentheses e to the power of 2 x end exponent times 2 times ln x plus e to the power of 2 x end exponent times 1 over x close parentheses equals 3 times e to the power of 2 x end exponent times open parentheses 2 ln x plus 1 over x close parentheses

      c5/4c/1333694f249760b916894e02b963.png” alt=”times fraction numerator 2 cos x times open parentheses cos x close parentheses apostrophe times open parentheses x cubed minus 3 x close parentheses minus cos squared x times open parentheses 3 x squared minus 3 close parentheses over denominator open parentheses x cubed minus 3 x close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#183;«/mo»«mfrac»«mrow»«mn»2«/mn»«mi»cos«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»`«/mo»«mo»§#183;«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mo»-«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      1 fifth times fifth root of open parentheses fraction numerator x cubed minus 3 x over denominator cos squared x end fraction close parentheses to the power of 4 end root times fraction numerator 2 cos x times open parentheses negative sin x close parentheses times open parentheses x cubed minus 3 x close parentheses minus cos squared x times open parentheses 3 x squared minus 3 close parentheses over denominator open parentheses x cubed minus 3 x close parentheses squared end fraction equals

      1 fifth times fraction numerator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 4 end root over denominator fifth root of open parentheses cos squared x close parentheses to the power of 4 end root end fraction times fraction numerator negative cos x times open square brackets 2 sin x times open parentheses x cubed minus 3 x close parentheses plus cos x times open parentheses 3 x squared minus 3 close parentheses close square brackets over denominator fifth root of open parentheses open parentheses x cubed minus 3 x close parentheses squared close parentheses to the power of 5 end root end fraction equals

      equals negative 1 fifth times fraction numerator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 4 end root over denominator fifth root of cos to the power of 8 x end root end fraction times fraction numerator fifth root of cos to the power of 5 x end root times open square brackets 2 times open parentheses x cubed minus 3 x close parentheses times sin x plus open parentheses 3 x squared minus 3 close parentheses times cos x close square brackets over denominator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 10 end root end fraction equals

      equals negative 1 fifth times fraction numerator 2 times open parentheses x cubed minus 3 x close parentheses times sin x plus open parentheses 3 x squared minus 3 close parentheses times cos x over denominator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 6 times cos cubed x end root end fraction

      dc/9d/43f65dd64634765de9142c72d807.png” alt=”4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«mfrac»«mrow»«mo»-«/mo»«mn»25«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»-«/mo»«mn»10«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«/mrow»«mrow»«msup»«mi»x«/mi»«mstyle displaystyle=¨true¨»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mstyle»«/msup»«mo»§#183;«/mo»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/math»” />

       

      02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  25. Leszek pisze:

     Witam,nie wiem czy kalkulator dobrze liczy ale wychodzi że (ln(x))’ = 1/x i to jest dobrze ale wpisując ln(2x) podaje wynik też 1/x czy to jest aby dobrze? Czy nie powinno być 2/x ?Proszę o szybką odpowiedź.

    1. Tutaj wynik jest poprawny, pochodna open parentheses ln left parenthesis 2 x right parenthesis close parentheses apostrophe equals 1 over x
      Bierze się to z tego, że jest to złożenie dwóch funkcji  – nie ma Pan samego “x” w logarytmie tylko coś więcej. Przy liczeniu takich pochodnych, najpierw robimy pochodną tej funkcji “zewnętrznej” i domnażamy do niej pochodną funkcji w środku, tej “wewnętrznej”. 

      Ogólnie na wzorach to idzie tak: left parenthesis f left parenthesis g left parenthesis x right parenthesis right parenthesis apostrophe space equals space f apostrophe left parenthesis g left parenthesis x right parenthesis right parenthesis space times space g apostrophe left parenthesis x right parenthesis

      Przy naszych danych to pójdzie tak: open parentheses ln left parenthesis increment right parenthesis close parentheses apostrophe equals 1 over increment times increment apostrophe  , gdzie za ten increment biorę funkcję wewnętrzną.

      Stąd ostatecznie: open parentheses ln left parenthesis 2 x right parenthesis close parentheses apostrophe equals fraction numerator 1 over denominator 2 x end fraction times open parentheses 2 x close parentheses apostrophe equals fraction numerator 1 over denominator 2 x end fraction times 2 times 1 equals fraction numerator 2 over denominator 2 x end fraction equals 1 over x

      56/59/da565d75ad307a420ee679d5b107.png” alt=”a r c sin x plus a r c cos x equals C equals a r c sin 0 plus a r c cos 0 equals 0 plus pi over 2 equals pi over 2″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»=«/mo»«mi»C«/mi»«mo»=«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mn»0«/mn»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mn»0«/mn»«mo»=«/mo»«mn»0«/mn»«mo»+«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«/math»” />

      Wtedy funkcja

      y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r c sin x plus a r c cos x close vertical bar end root equals fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 5,

      i jej pochodna

      (wg wzoru dla funkcji złożonej:  open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 triangle to the power of 4 times open parentheses triangle close parentheses apostrophe   )

      wynosi:

      y apostrophe equals fifth root of pi over 2 end root times 5 times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses sin x plus cos x close parentheses apostrophe equals

      fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses

       

      b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

       

      02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  26. Michał pisze:

    Witam mógłby mi ktoś pomóc obliczyć pochodną funkcji y=cube root of x to the power of 5 end root ln xb0/ff/b22170a78fe40f0b6c585934d62f.png” alt=”x squared minus x minus 2 not equal to 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«mo»§#8800;«/mo»«mn»0«/mn»«/math»”>
    capital delta equals left parenthesis negative 1 right parenthesis squared minus 4 times 1 times left parenthesis negative 2 right parenthesis equals 9
    x subscript 1 equals fraction numerator 1 minus square root of 9 over denominator 2 end fraction equals fraction numerator 1 minus 3 over denominator 2 end fraction equals fraction numerator negative 2 over denominator 2 end fraction equals negative 1
    space x subscript 2 equals fraction numerator 1 plus square root of 9 over denominator 2 end fraction equals fraction numerator 1 plus 3 over denominator 2 end fraction equals 4 over 2 equals 2

    Zatem D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

    Przechodzimy do wyznaczania monotoniczności funkcji f. W tym celu obliczymy jej pochodną i sprawdzimy, kiedy jest dodatnia, a kiedy ujemna.

    f apostrophe left parenthesis x right parenthesis equals fraction numerator open parentheses x cubed close parentheses apostrophe times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses x squared minus x minus 2 close parentheses apostrophe over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals
    equals fraction numerator 3 x squared times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses 2 x minus 1 close parentheses over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals fraction numerator 3 x to the power of 4 minus 3 x cubed minus 6 x squared minus 2 x to the power of 4 plus x cubed over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals
    equals fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction

    Zbadamy teraz, kiedy pochodna przyjmuje wartości większe lub równe 0, a kiedy mniejsze lub równe 0.

    fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0

    x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

    x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

    capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
    x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
    x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
    wykres

    Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
    Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

     

    Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

     

    Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

    1. y equals cube root of x to the power of 5 end root times ln x equals x to the power of 5 over 3 end exponent times ln x

      Stosuję wzór: open parentheses u times v close parentheses apostrophe equals u apostrophe times v plus u times v apostrophe

      y apostrophe equals open parentheses x to the power of 5 over 3 end exponent close parentheses apostrophe times ln x plus x to the power of 5 over 3 end exponent times open parentheses ln x close parentheses apostrophe equals 5 over 3 times x to the power of 5 over 3 minus 1 end exponent times ln x plus x to the power of 5 over 3 end exponent times 1 over x equals

      5 over 3 times x to the power of 2 over 3 end exponent times ln x plus x to the power of begin display style 5 over 3 end style end exponent over x equals 5 over 3 times x to the power of 2 over 3 end exponent times ln x plus x to the power of 5 over 3 minus 1 end exponent equals 5 over 3 times x to the power of 2 over 3 end exponent times ln x plus x to the power of 2 over 3 end exponent equals

      x to the power of 2 over 3 end exponent times open parentheses 5 over 3 ln x plus 1 close parentheses equals cube root of x squared end root times open parentheses 5 over 3 ln x plus 1 close parentheses

      ce/50/e97f118f5da367c319b5e5b5e657.png” alt=”3 times open parentheses e to the power of 2 x end exponent times 2 times ln x plus e to the power of 2 x end exponent times 1 over x close parentheses equals 3 times e to the power of 2 x end exponent times open parentheses 2 ln x plus 1 over x close parentheses” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mo»§#183;«/mo»«mfenced»«mrow»«msup»«mi»e«/mi»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«/msup»«mo»§#183;«/mo»«mn»2«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»+«/mo»«msup»«mi»e«/mi»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«mo»=«/mo»«mn»3«/mn»«mo»§#183;«/mo»«msup»«mi»e«/mi»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«/msup»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi»ln«/mi»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«/math»” />c5/4c/1333694f249760b916894e02b963.png” alt=”times fraction numerator 2 cos x times open parentheses cos x close parentheses apostrophe times open parentheses x cubed minus 3 x close parentheses minus cos squared x times open parentheses 3 x squared minus 3 close parentheses over denominator open parentheses x cubed minus 3 x close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#183;«/mo»«mfrac»«mrow»«mn»2«/mn»«mi»cos«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»`«/mo»«mo»§#183;«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mo»-«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      1 fifth times fifth root of open parentheses fraction numerator x cubed minus 3 x over denominator cos squared x end fraction close parentheses to the power of 4 end root times fraction numerator 2 cos x times open parentheses negative sin x close parentheses times open parentheses x cubed minus 3 x close parentheses minus cos squared x times open parentheses 3 x squared minus 3 close parentheses over denominator open parentheses x cubed minus 3 x close parentheses squared end fraction equals

      1 fifth times fraction numerator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 4 end root over denominator fifth root of open parentheses cos squared x close parentheses to the power of 4 end root end fraction times fraction numerator negative cos x times open square brackets 2 sin x times open parentheses x cubed minus 3 x close parentheses plus cos x times open parentheses 3 x squared minus 3 close parentheses close square brackets over denominator fifth root of open parentheses open parentheses x cubed minus 3 x close parentheses squared close parentheses to the power of 5 end root end fraction equals

      equals negative 1 fifth times fraction numerator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 4 end root over denominator fifth root of cos to the power of 8 x end root end fraction times fraction numerator fifth root of cos to the power of 5 x end root times open square brackets 2 times open parentheses x cubed minus 3 x close parentheses times sin x plus open parentheses 3 x squared minus 3 close parentheses times cos x close square brackets over denominator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 10 end root end fraction equals

      equals negative 1 fifth times fraction numerator 2 times open parentheses x cubed minus 3 x close parentheses times sin x plus open parentheses 3 x squared minus 3 close parentheses times cos x over denominator fifth root of open parentheses x cubed minus 3 x close parentheses to the power of 6 times cos cubed x end root end fraction

      dc/9d/43f65dd64634765de9142c72d807.png” alt=”4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«mfrac»«mrow»«mo»-«/mo»«mn»25«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»-«/mo»«mn»10«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«/mrow»«mrow»«msup»«mi»x«/mi»«mstyle displaystyle=¨true¨»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mstyle»«/msup»«mo»§#183;«/mo»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/math»” />

       

      02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  27. marco pisze:

    jak obliczyć pochodna funkcjiy= 4x^1/3 * (5 + 2*4^3^x)/x^2 + 1

    1. y equals fraction numerator 4 cube root of x times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses over denominator x squared plus 1 end fraction equals 4 times fraction numerator cube root of x times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses over denominator x squared plus 1 end fraction

      Stosuję wzory:

      open parentheses u over v close parentheses apostrophe equals fraction numerator u apostrophe v minus u v apostrophe over denominator v squared end fraction oraz (pochodna funkcji złożonej) open parentheses 4 to the power of triangle close parentheses apostrophe equals 4 to the power of triangle times ln 4 times open parentheses triangle close parentheses apostrophe, a także

      open parentheses u times v close parentheses apostrophe equals u apostrophe v plus u v apostrophe

      y apostrophe equals 4 times fraction numerator open square brackets cube root of x times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses close square brackets apostrophe times open parentheses x squared plus 1 close parentheses minus cube root of x times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses times open parentheses x squared plus 1 close parentheses apostrophe over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals

      4 times fraction numerator open square brackets open parentheses cube root of x close parentheses apostrophe times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses plus cube root of x times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses apostrophe close square brackets times open parentheses x squared plus 1 close parentheses minus cube root of x times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses times open parentheses 2 x plus 0 close parentheses over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals

      4 times fraction numerator open square brackets open parentheses x to the power of begin display style 1 third end style end exponent close parentheses apostrophe times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses plus cube root of x times open parentheses 0 plus 2 times 4 to the power of 3 to the power of x end exponent times ln 4 times open parentheses 3 to the power of x close parentheses apostrophe close parentheses close square brackets times open parentheses x squared plus 1 close parentheses minus cube root of x times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses times 2 x over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals

      4 times fraction numerator open square brackets begin display style 1 third end style x to the power of negative begin display style 2 over 3 end style end exponent times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses plus x to the power of begin display style 1 third end style end exponent times 2 times 4 to the power of 3 to the power of x end exponent times ln 4 times 3 to the power of x times ln 3 close square brackets times open parentheses x squared plus 1 close parentheses minus x to the power of begin display style 1 third end style end exponent times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses times 2 x over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals

      4 times fraction numerator begin display style 1 third end style x to the power of negative begin display style 2 over 3 end style end exponent times open square brackets 5 plus 2 times 4 to the power of 3 to the power of x end exponent plus 3 x times 2 times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 close square brackets times open parentheses x squared plus 1 close parentheses minus begin display style 1 third end style x to the power of negative begin display style 2 over 3 end style end exponent times 3 x times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses times 2 x over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals

      4 times 1 third x to the power of negative 2 over 3 end exponent times fraction numerator open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 close parentheses times open parentheses x squared plus 1 close parentheses minus 6 x squared times open parentheses 5 plus 2 times 4 to the power of 3 to the power of x end exponent close parentheses over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals

      4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals

      4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

       

      02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  28. Klaudia pisze:

    Witam, jak obliczyć pochodną funkcji f(x)= e^(2x+1)/(x-2)

  29. Sylwia pisze:

    Witam, czy mógłby mi Pan wytłumaczyć jak rozwiązać taką pochodną:f(x)=xsinxlnx ?z góry dziękuje i pozdrawiam 🙂

    1. f open parentheses x close parentheses equals x times sin x times ln x

      Znany jest wzór dla pochodnej iloczynu:

      open parentheses u times v close parentheses apostrophe equals u apostrophe v plus u v apostrophe

      Spróbujemy otrzymać wzór dla iloczynu trzech czynników:

      open parentheses u times v times w close parentheses apostrophe equals open square brackets u times open parentheses v times w close parentheses close square brackets apostrophe equals u apostrophe times open parentheses v times w close parentheses plus u times open parentheses v times w close parentheses apostrophe equals u apostrophe times v times w plus u times open parentheses v apostrophe times w plus v times w apostrophe close parentheses equals

      u apostrophe times v times w plus u times v apostrophe times w plus u times v times w apostrophe

      Wtedy:

      f apostrophe open parentheses x close parentheses equals open parentheses x times sin x times ln x close parentheses apostrophe equals x apostrophe times sin x times ln x plus x times open parentheses sin x close parentheses apostrophe times ln x plus x times sin x times open parentheses ln x close parentheses apostrophe equals

      1 times sin x times ln x plus x times cos x times ln x plus x times sin x times 1 over x equals sin x times ln x plus x times cos x times ln x plus sin x

      94/80/0f49a083125dc60389915049c352.png” alt=”f open parentheses x close parentheses equals x squared times open parentheses x minus 2 close parentheses squared equals open square brackets x times open parentheses x minus 2 close parentheses close square brackets squared equals open parentheses x squared minus 2 x close parentheses squared equals x to the power of 4 minus 4 x cubed plus 4 x squared” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/math»” />

      Wtedy:

      f apostrophe open parentheses x close parentheses equals open parentheses x to the power of 4 minus 4 x cubed plus 4 x squared close parentheses apostrophe equals 4 x cubed minus 4 times 3 x squared plus 4 times 2 x equals 4 x cubed minus 12 x squared plus 8 x

      ad/34/1a7bdcfe3c5aeadd58e56de14e57.png” alt=”equals fraction numerator 3 x squared times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses 2 x minus 1 close parentheses over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals fraction numerator 3 x to the power of 4 minus 3 x cubed minus 6 x squared minus 2 x to the power of 4 plus x cubed over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»-«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />
      equals fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction

      Zbadamy teraz, kiedy pochodna przyjmuje wartości większe lub równe 0, a kiedy mniejsze lub równe 0.

      fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  30. radek pisze:

    (2x-1)^4=8(2x-1)^3 dlaczego tak????

  31. Jack pisze:

    Jak to obliczyć ? f(x)= open parentheses fraction numerator 1 over denominator 2 square root of x end fraction minus 1 close parentheses x squaredb0/ff/b22170a78fe40f0b6c585934d62f.png” alt=”x squared minus x minus 2 not equal to 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«mo»§#8800;«/mo»«mn»0«/mn»«/math»”>
    capital delta equals left parenthesis negative 1 right parenthesis squared minus 4 times 1 times left parenthesis negative 2 right parenthesis equals 9
    x subscript 1 equals fraction numerator 1 minus square root of 9 over denominator 2 end fraction equals fraction numerator 1 minus 3 over denominator 2 end fraction equals fraction numerator negative 2 over denominator 2 end fraction equals negative 1
    space x subscript 2 equals fraction numerator 1 plus square root of 9 over denominator 2 end fraction equals fraction numerator 1 plus 3 over denominator 2 end fraction equals 4 over 2 equals 2

    Zatem D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

    Przechodzimy do wyznaczania monotoniczności funkcji f. W tym celu obliczymy jej pochodną i sprawdzimy, kiedy jest dodatnia, a kiedy ujemna.

    f apostrophe left parenthesis x right parenthesis equals fraction numerator open parentheses x cubed close parentheses apostrophe times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses x squared minus x minus 2 close parentheses apostrophe over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals
    equals fraction numerator 3 x squared times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses 2 x minus 1 close parentheses over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals fraction numerator 3 x to the power of 4 minus 3 x cubed minus 6 x squared minus 2 x to the power of 4 plus x cubed over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals
    equals fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction

    Zbadamy teraz, kiedy pochodna przyjmuje wartości większe lub równe 0, a kiedy mniejsze lub równe 0.

    fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0

    x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

    x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

    capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
    x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
    x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
    wykres

    Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
    Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

     

    Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

     

    Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  32. Iza pisze:

    f(x)=x^2*(x-2)^2Wytłumaczysz mi jak to policzyłeś, trochę inaczej mam rozpisane z zajęć i się pogubiłam…? Z góry dziękuję 🙂

    1. f open parentheses x close parentheses equals x squared times open parentheses x minus 2 close parentheses squared

      Stosuję wzór:

      open parentheses u times v close parentheses apostrophe equals u apostrophe v plus u v apostrophe oraz (pochodna funkcji złożonej) open parentheses triangle squared close parentheses apostrophe equals 2 triangle times open parentheses triangle close parentheses apostrophe

      f apostrophe open parentheses x close parentheses equals open square brackets x squared times open parentheses x minus 2 close parentheses squared close square brackets apostrophe equals open parentheses x squared close parentheses apostrophe times open parentheses x minus 2 close parentheses squared plus x squared times open square brackets open parentheses x minus 2 close parentheses squared close square brackets apostrophe equals 2 x times open parentheses x minus 2 close parentheses squared plus

      plus x squared times 2 times open parentheses x minus 2 close parentheses times open parentheses x minus 2 close parentheses apostrophe equals 2 x times open parentheses x squared minus 4 x plus 4 close parentheses plus 2 x squared times open parentheses x minus 2 close parentheses times open parentheses 1 minus 0 close parentheses equals

      2 x cubed minus 8 x squared plus 8 x plus 2 x cubed minus 4 x squared equals 4 x cubed minus 12 x squared plus 8 x

      Można było inaczej:

      f open parentheses x close parentheses equals x squared times open parentheses x minus 2 close parentheses squared equals open square brackets x times open parentheses x minus 2 close parentheses close square brackets squared equals open parentheses x squared minus 2 x close parentheses squared equals x to the power of 4 minus 4 x cubed plus 4 x squared

      Wtedy:

      f apostrophe open parentheses x close parentheses equals open parentheses x to the power of 4 minus 4 x cubed plus 4 x squared close parentheses apostrophe equals 4 x cubed minus 4 times 3 x squared plus 4 times 2 x equals 4 x cubed minus 12 x squared plus 8 x

      ad/34/1a7bdcfe3c5aeadd58e56de14e57.png” alt=”equals fraction numerator 3 x squared times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses 2 x minus 1 close parentheses over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals fraction numerator 3 x to the power of 4 minus 3 x cubed minus 6 x squared minus 2 x to the power of 4 plus x cubed over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»-«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />
      equals fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction

      Zbadamy teraz, kiedy pochodna przyjmuje wartości większe lub równe 0, a kiedy mniejsze lub równe 0.

      fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  33. michalaczek pisze:

    f(x)=lnfifth root of efb/07/e160807668a4b91383f2c6d245b0.png” alt=”ln to the power of 8 x” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»ln«/mi»«mn»8«/mn»«/msup»«mi»x«/mi»«/math»” />3) f(x)=ln(5 x to the power of 4 minus x plus 9)be/e0/30079ee2ce183436018ad1559d9d.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe apostrophe end cell equals cell open parentheses fraction numerator begin display style 2 minus ln x end style over denominator 2 x square root of x end fraction close parentheses to the power of apostrophe equals fraction numerator begin display style open parentheses 2 minus ln x close parentheses apostrophe times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x square root of x close parentheses apostrophe end style over denominator open parentheses 2 x square root of x close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x to the power of bevelled 3 over 2 end exponent close parentheses apostrophe end style over denominator open parentheses 2 x to the power of begin display style bevelled 3 over 2 end style end exponent close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times 2 times 3 over 2 x to the power of bevelled 1 half end exponent end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus open parentheses 2 minus ln x close parentheses times 3 square root of x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus 6 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 8 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style square root of x open parentheses 3 minus 8 ln x close parentheses end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 8 ln x end style over denominator 4 x squared square root of x end fraction end cell end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«mo»`«/mo»«mo»`«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mfrac»«mstyle displaystyle=¨true¨»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mfenced»«mo»`«/mo»«/msup»«mo»=«/mo»«mfrac»«mstyle displaystyle=¨true¨»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»`«/mo»«mo»§#183;«/mo»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfenced»«mo»`«/mo»«/mstyle»«msup»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mfenced»«mrow»«mo»-«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mfrac bevelled=¨true¨»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mrow»«/mfenced»«mo»`«/mo»«/mstyle»«msup»«mfenced»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mstyle displaystyle=¨true¨»«mfrac bevelled=¨true¨»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/mstyle»«/msup»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mfenced»«mrow»«mo»-«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»2«/mn»«mo»§#183;«/mo»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«msup»«mi»x«/mi»«mfrac bevelled=¨true¨»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mo»-«/mo»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»3«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mo»-«/mo»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mn»6«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»+«/mo»«mn»3«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mo»-«/mo»«mn»8«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»+«/mo»«mn»3«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«msqrt»«mi»x«/mi»«/msqrt»«mfenced»«mrow»«mn»3«/mn»«mo»-«/mo»«mn»8«/mn»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mn»3«/mn»«mo»-«/mo»«mn»8«/mn»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” />2a/00/aff50b4236992db275aec9e9aa54.png” alt=”equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times open parentheses negative begin display style 1 over x squared end style plus begin display style 1 over x end style close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«mo»§#183;«/mo»«mfenced»«mrow»«mo»-«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mstyle»«mo»+«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»-«/mo»«mfenced»«mrow»«mn»12«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§#183;«/mo»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«msup»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times begin display style fraction numerator negative 1 plus x over denominator x squared end fraction end style times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals

    equals fraction numerator open parentheses x minus 1 close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus 2 x squared times open parentheses 12 x cubed plus 3 x squared close parentheses times open parentheses begin display style 1 over x end style plus ln x close parentheses over denominator 2 x squared times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

    equals fraction numerator 3 x to the power of 5 plus x to the power of 4 plus x minus 3 x to the power of 4 minus x cubed minus 1 minus 24 x to the power of 4 minus 6 x cubed minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 open parentheses x times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

    equals fraction numerator 3 x to the power of 5 minus 26 x to the power of 4 minus 7 x cubed plus x minus 1 minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 times open parentheses 3 x to the power of 5 plus x to the power of 4 plus x close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction

    b0/89/71867873aebda1ce5319045f8f36.png” alt=”left right double arrow” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8660;«/mo»«/math»” />: taka obustronna strzałka jest również spójnikiem logicznym i oznacza tyle, co “wtedy i tylko wtedy”. Z lewej strony wynika prawa, a z prawej lewa. Przykład: x equals 2 space logical or space x equals negative 2 space left right double arrow space x squared equals 4 

    logical or: alternatywa – kolejny spójnik logiczny, odpowiednik słowa “lub”

    logical and: koniunkcja – spójnik logiczny, odpowiednik słowa “i”

    tilde: negacja – spójnik logiczny oznaczający zaprzeczenie

    for all: kwantyfikator duży, ogólny oznaczający “dla każdego … zachodzi …”

    there exists: kwantyfikator mały, szczegółowy oznaczający “istnieje …, takie że …”

    Więcej informacji o spójnikach logicznych można znaleźć w lekcji:
    https://online.etrapez.pl/lesson/lekcja-2-tabele-i-spojniki-logiczne-przypisywanie-wartosci-zdaniom-zlozonym/

     

    Więcej informacji o kwantyfikatorach można znaleźć w lekcji:
    https://online.etrapez.pl/wybor-kursu/matematyka-dyskretna/lekcja-29-kwantyfikatory/

    1. michalaczek pisze:

      czy moge zapisac w postaci log subscript e e to the power of 5 equals 5  ???fb/07/e160807668a4b91383f2c6d245b0.png” alt=”ln to the power of 8 x” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»ln«/mi»«mn»8«/mn»«/msup»«mi»x«/mi»«/math»” />3) f(x)=ln(5 x to the power of 4 minus x plus 9)be/e0/30079ee2ce183436018ad1559d9d.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe apostrophe end cell equals cell open parentheses fraction numerator begin display style 2 minus ln x end style over denominator 2 x square root of x end fraction close parentheses to the power of apostrophe equals fraction numerator begin display style open parentheses 2 minus ln x close parentheses apostrophe times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x square root of x close parentheses apostrophe end style over denominator open parentheses 2 x square root of x close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x to the power of bevelled 3 over 2 end exponent close parentheses apostrophe end style over denominator open parentheses 2 x to the power of begin display style bevelled 3 over 2 end style end exponent close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times 2 times 3 over 2 x to the power of bevelled 1 half end exponent end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus open parentheses 2 minus ln x close parentheses times 3 square root of x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus 6 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 8 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style square root of x open parentheses 3 minus 8 ln x close parentheses end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 8 ln x end style over denominator 4 x squared square root of x end fraction end cell end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«mo»`«/mo»«mo»`«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mfrac»«mstyle displaystyle=¨true¨»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mfenced»«mo»`«/mo»«/msup»«mo»=«/mo»«mfrac»«mstyle displaystyle=¨true¨»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»`«/mo»«mo»§#183;«/mo»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfenced»«mo»`«/mo»«/mstyle»«msup»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mfenced»«mrow»«mo»-«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mfrac bevelled=¨true¨»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mrow»«/mfenced»«mo»`«/mo»«/mstyle»«msup»«mfenced»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mstyle displaystyle=¨true¨»«mfrac bevelled=¨true¨»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/mstyle»«/msup»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mfenced»«mrow»«mo»-«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»2«/mn»«mo»§#183;«/mo»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«msup»«mi»x«/mi»«mfrac bevelled=¨true¨»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mo»-«/mo»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»3«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mo»-«/mo»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mn»6«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»+«/mo»«mn»3«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mo»-«/mo»«mn»8«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»+«/mo»«mn»3«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«msqrt»«mi»x«/mi»«/msqrt»«mfenced»«mrow»«mn»3«/mn»«mo»-«/mo»«mn»8«/mn»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mn»3«/mn»«mo»-«/mo»«mn»8«/mn»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” />2a/00/aff50b4236992db275aec9e9aa54.png” alt=”equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times open parentheses negative begin display style 1 over x squared end style plus begin display style 1 over x end style close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«mo»§#183;«/mo»«mfenced»«mrow»«mo»-«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mstyle»«mo»+«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»-«/mo»«mfenced»«mrow»«mn»12«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§#183;«/mo»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«msup»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times begin display style fraction numerator negative 1 plus x over denominator x squared end fraction end style times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals

      equals fraction numerator open parentheses x minus 1 close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus 2 x squared times open parentheses 12 x cubed plus 3 x squared close parentheses times open parentheses begin display style 1 over x end style plus ln x close parentheses over denominator 2 x squared times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

      equals fraction numerator 3 x to the power of 5 plus x to the power of 4 plus x minus 3 x to the power of 4 minus x cubed minus 1 minus 24 x to the power of 4 minus 6 x cubed minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 open parentheses x times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

      equals fraction numerator 3 x to the power of 5 minus 26 x to the power of 4 minus 7 x cubed plus x minus 1 minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 times open parentheses 3 x to the power of 5 plus x to the power of 4 plus x close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction

      b0/89/71867873aebda1ce5319045f8f36.png” alt=”left right double arrow” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8660;«/mo»«/math»” />: taka obustronna strzałka jest również spójnikiem logicznym i oznacza tyle, co “wtedy i tylko wtedy”. Z lewej strony wynika prawa, a z prawej lewa. Przykład: x equals 2 space logical or space x equals negative 2 space left right double arrow space x squared equals 4 

      logical or: alternatywa – kolejny spójnik logiczny, odpowiednik słowa “lub”

      logical and: koniunkcja – spójnik logiczny, odpowiednik słowa “i”

      tilde: negacja – spójnik logiczny oznaczający zaprzeczenie

      for all: kwantyfikator duży, ogólny oznaczający “dla każdego … zachodzi …”

      there exists: kwantyfikator mały, szczegółowy oznaczający “istnieje …, takie że …”

      Więcej informacji o spójnikach logicznych można znaleźć w lekcji:
      https://online.etrapez.pl/lesson/lekcja-2-tabele-i-spojniki-logiczne-przypisywanie-wartosci-zdaniom-zlozonym/

       

      Więcej informacji o kwantyfikatorach można znaleźć w lekcji:
      https://online.etrapez.pl/wybor-kursu/matematyka-dyskretna/lekcja-29-kwantyfikatory/

  34. michalaczek pisze:

    1) f(x)=e to the power of negative x end exponent2) f(x)=ln to the power of 8 x3) f(x)=ln(5 x to the power of 4 minus x plus 9)be/e0/30079ee2ce183436018ad1559d9d.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe apostrophe end cell equals cell open parentheses fraction numerator begin display style 2 minus ln x end style over denominator 2 x square root of x end fraction close parentheses to the power of apostrophe equals fraction numerator begin display style open parentheses 2 minus ln x close parentheses apostrophe times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x square root of x close parentheses apostrophe end style over denominator open parentheses 2 x square root of x close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x to the power of bevelled 3 over 2 end exponent close parentheses apostrophe end style over denominator open parentheses 2 x to the power of begin display style bevelled 3 over 2 end style end exponent close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times 2 times 3 over 2 x to the power of bevelled 1 half end exponent end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus open parentheses 2 minus ln x close parentheses times 3 square root of x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus 6 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 8 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style square root of x open parentheses 3 minus 8 ln x close parentheses end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 8 ln x end style over denominator 4 x squared square root of x end fraction end cell end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«mo»`«/mo»«mo»`«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mfrac»«mstyle displaystyle=¨true¨»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mfenced»«mo»`«/mo»«/msup»«mo»=«/mo»«mfrac»«mstyle displaystyle=¨true¨»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»`«/mo»«mo»§#183;«/mo»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfenced»«mo»`«/mo»«/mstyle»«msup»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mfenced»«mrow»«mo»-«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mfrac bevelled=¨true¨»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mrow»«/mfenced»«mo»`«/mo»«/mstyle»«msup»«mfenced»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mstyle displaystyle=¨true¨»«mfrac bevelled=¨true¨»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/mstyle»«/msup»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mfenced»«mrow»«mo»-«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»2«/mn»«mo»§#183;«/mo»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«msup»«mi»x«/mi»«mfrac bevelled=¨true¨»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mo»-«/mo»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»3«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mo»-«/mo»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mn»6«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»+«/mo»«mn»3«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mo»-«/mo»«mn»8«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»+«/mo»«mn»3«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«msqrt»«mi»x«/mi»«/msqrt»«mfenced»«mrow»«mn»3«/mn»«mo»-«/mo»«mn»8«/mn»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mn»3«/mn»«mo»-«/mo»«mn»8«/mn»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” />2a/00/aff50b4236992db275aec9e9aa54.png” alt=”equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times open parentheses negative begin display style 1 over x squared end style plus begin display style 1 over x end style close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«mo»§#183;«/mo»«mfenced»«mrow»«mo»-«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mstyle»«mo»+«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»-«/mo»«mfenced»«mrow»«mn»12«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§#183;«/mo»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«msup»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times begin display style fraction numerator negative 1 plus x over denominator x squared end fraction end style times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals

    equals fraction numerator open parentheses x minus 1 close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus 2 x squared times open parentheses 12 x cubed plus 3 x squared close parentheses times open parentheses begin display style 1 over x end style plus ln x close parentheses over denominator 2 x squared times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

    equals fraction numerator 3 x to the power of 5 plus x to the power of 4 plus x minus 3 x to the power of 4 minus x cubed minus 1 minus 24 x to the power of 4 minus 6 x cubed minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 open parentheses x times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

    equals fraction numerator 3 x to the power of 5 minus 26 x to the power of 4 minus 7 x cubed plus x minus 1 minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 times open parentheses 3 x to the power of 5 plus x to the power of 4 plus x close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction

    b0/89/71867873aebda1ce5319045f8f36.png” alt=”left right double arrow” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8660;«/mo»«/math»” />: taka obustronna strzałka jest również spójnikiem logicznym i oznacza tyle, co “wtedy i tylko wtedy”. Z lewej strony wynika prawa, a z prawej lewa. Przykład: x equals 2 space logical or space x equals negative 2 space left right double arrow space x squared equals 4 

    logical or: alternatywa – kolejny spójnik logiczny, odpowiednik słowa “lub”

    logical and: koniunkcja – spójnik logiczny, odpowiednik słowa “i”

    tilde: negacja – spójnik logiczny oznaczający zaprzeczenie

    for all: kwantyfikator duży, ogólny oznaczający “dla każdego … zachodzi …”

    there exists: kwantyfikator mały, szczegółowy oznaczający “istnieje …, takie że …”

    Więcej informacji o spójnikach logicznych można znaleźć w lekcji:
    https://online.etrapez.pl/lesson/lekcja-2-tabele-i-spojniki-logiczne-przypisywanie-wartosci-zdaniom-zlozonym/

     

    Więcej informacji o kwantyfikatorach można znaleźć w lekcji:
    https://online.etrapez.pl/wybor-kursu/matematyka-dyskretna/lekcja-29-kwantyfikatory/

  35. michalaczek pisze:

    odp do f(x)=square root of 4 x to the power of 7 plus 1 end root  to  fraction numerator 14 x cubed over denominator square root of 4 x to the power of 7 plus 1 end root end fraction   ???5f/e2/3224c3d50cd18d14b127ab76e98c.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe end cell equals cell fraction numerator open parentheses ln x close parentheses apostrophe times square root of x minus ln x times open parentheses square root of x close parentheses apostrophe over denominator open parentheses square root of x close parentheses squared end fraction equals fraction numerator begin display style 1 over x end style times square root of x minus ln x times begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style over denominator x end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator square root of x over denominator x end fraction minus fraction numerator ln x over denominator 2 square root of x end fraction end style over denominator x end fraction times fraction numerator 2 square root of x over denominator 2 square root of x end fraction equals fraction numerator begin display style fraction numerator 2 x over denominator x end fraction minus ln x end style over denominator 2 x square root of x end fraction end cell row blank equals cell fraction numerator begin display style 2 minus ln x end style over denominator 2 x square root of x end fraction end cell end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«mo»`«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mfenced»«mrow»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»`«/mo»«mo»§#183;«/mo»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«msqrt»«mi»x«/mi»«/msqrt»«/mfenced»«mo»`«/mo»«/mrow»«msup»«mfenced»«msqrt»«mi»x«/mi»«/msqrt»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»§#183;«/mo»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«/mrow»«mi»x«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mfrac»«msqrt»«mi»x«/mi»«/msqrt»«mi»x«/mi»«/mfrac»«mo»-«/mo»«mfrac»«mrow»«mi»ln«/mi»«mi»x«/mi»«/mrow»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«mi»x«/mi»«/mfrac»«mo»§#183;«/mo»«mfrac»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mstyle displaystyle=¨true¨»«mfrac»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«mi»x«/mi»«/mfrac»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” />

    I druga pochodna

    table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe apostrophe end cell equals cell open parentheses fraction numerator begin display style 2 minus ln x end style over denominator 2 x square root of x end fraction close parentheses to the power of apostrophe equals fraction numerator begin display style open parentheses 2 minus ln x close parentheses apostrophe times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x square root of x close parentheses apostrophe end style over denominator open parentheses 2 x square root of x close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x to the power of bevelled 3 over 2 end exponent close parentheses apostrophe end style over denominator open parentheses 2 x to the power of begin display style bevelled 3 over 2 end style end exponent close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times 2 times 3 over 2 x to the power of bevelled 1 half end exponent end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus open parentheses 2 minus ln x close parentheses times 3 square root of x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus 6 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 8 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style square root of x open parentheses 3 minus 8 ln x close parentheses end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 8 ln x end style over denominator 4 x squared square root of x end fraction end cell end table2a/00/aff50b4236992db275aec9e9aa54.png” alt=”equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times open parentheses negative begin display style 1 over x squared end style plus begin display style 1 over x end style close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«mo»§#183;«/mo»«mfenced»«mrow»«mo»-«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mstyle»«mo»+«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»-«/mo»«mfenced»«mrow»«mn»12«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§#183;«/mo»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«msup»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times begin display style fraction numerator negative 1 plus x over denominator x squared end fraction end style times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals

    equals fraction numerator open parentheses x minus 1 close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus 2 x squared times open parentheses 12 x cubed plus 3 x squared close parentheses times open parentheses begin display style 1 over x end style plus ln x close parentheses over denominator 2 x squared times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

    equals fraction numerator 3 x to the power of 5 plus x to the power of 4 plus x minus 3 x to the power of 4 minus x cubed minus 1 minus 24 x to the power of 4 minus 6 x cubed minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 open parentheses x times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

    equals fraction numerator 3 x to the power of 5 minus 26 x to the power of 4 minus 7 x cubed plus x minus 1 minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 times open parentheses 3 x to the power of 5 plus x to the power of 4 plus x close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction

    b0/89/71867873aebda1ce5319045f8f36.png” alt=”left right double arrow” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8660;«/mo»«/math»” />: taka obustronna strzałka jest również spójnikiem logicznym i oznacza tyle, co “wtedy i tylko wtedy”. Z lewej strony wynika prawa, a z prawej lewa. Przykład: x equals 2 space logical or space x equals negative 2 space left right double arrow space x squared equals 4 

    logical or: alternatywa – kolejny spójnik logiczny, odpowiednik słowa “lub”

    logical and: koniunkcja – spójnik logiczny, odpowiednik słowa “i”

    tilde: negacja – spójnik logiczny oznaczający zaprzeczenie

    for all: kwantyfikator duży, ogólny oznaczający “dla każdego … zachodzi …”

    there exists: kwantyfikator mały, szczegółowy oznaczający “istnieje …, takie że …”

    Więcej informacji o spójnikach logicznych można znaleźć w lekcji:
    https://online.etrapez.pl/lesson/lekcja-2-tabele-i-spojniki-logiczne-przypisywanie-wartosci-zdaniom-zlozonym/

     

    Więcej informacji o kwantyfikatorach można znaleźć w lekcji:
    https://online.etrapez.pl/wybor-kursu/matematyka-dyskretna/lekcja-29-kwantyfikatory/

    1. f open parentheses x close parentheses equals square root of 4 x to the power of 7 plus 1 end root

      Stosuję wzór na pochodne funkcji złożonej:

      open parentheses square root of triangle close parentheses apostrophe equals fraction numerator 1 over denominator 2 square root of triangle end fraction times open parentheses triangle close parentheses apostrophe

      f apostrophe open parentheses x close parentheses equals fraction numerator 1 over denominator 2 square root of 4 x to the power of 7 plus 1 end root end fraction times open parentheses 4 x to the power of 7 plus 1 close parentheses apostrophe equals fraction numerator 4 times 7 x to the power of 6 plus 0 over denominator 2 square root of 4 x to the power of 7 plus 1 end root end fraction equals fraction numerator 28 x to the power of 6 over denominator 2 square root of 4 x to the power of 7 plus 1 end root end fraction equals fraction numerator 14 x to the power of 6 over denominator square root of 4 x to the power of 7 plus 1 end root end fraction

      06/0f/066dfe100a72f83b006d2302cca4.png” alt=”fraction numerator negative begin display style fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction end style times x minus a r c cos x times 1 over denominator x squared end fraction equals negative fraction numerator begin display style fraction numerator x over denominator square root of 1 minus x squared end root end fraction end style plus a r c cos x over denominator x squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mo»-«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«/mstyle»«mo»§#183;«/mo»«mi»x«/mi»«mo»-«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mn»1«/mn»«/mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«mo»-«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mi»x«/mi»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«/mstyle»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mi»x«/mi»«/mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      negative fraction numerator x plus square root of 1 minus x squared end root times a r c cos x over denominator x squared times square root of 1 minus x squared end root end fraction

      48/d2/4b6181539cac56b2709119d6442e.png” alt=”D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»D«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»§#8477;«/mi»«mo»«/mo»«mo»{«/mo»«mo»-«/mo»«mn»1«/mn»«mo»,«/mo»«mn»2«/mn»«mo»}«/mo»«/math»” />.

      Przechodzimy do wyznaczania monotoniczności funkcji f. W tym celu obliczymy jej pochodną i sprawdzimy, kiedy jest dodatnia, a kiedy ujemna.

      f apostrophe left parenthesis x right parenthesis equals fraction numerator open parentheses x cubed close parentheses apostrophe times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses x squared minus x minus 2 close parentheses apostrophe over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals
      equals fraction numerator 3 x squared times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses 2 x minus 1 close parentheses over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals fraction numerator 3 x to the power of 4 minus 3 x cubed minus 6 x squared minus 2 x to the power of 4 plus x cubed over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals
      equals fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction

      Zbadamy teraz, kiedy pochodna przyjmuje wartości większe lub równe 0, a kiedy mniejsze lub równe 0.

      fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  36. michalaczek pisze:

    f(x)=e to the power of 8 x end exponent     ???d8/b5/7af3651ec6f62fa9379c5edd24aa.png” alt=”bevelled fraction numerator a r c space sin space x over denominator x end fraction” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac bevelled=¨true¨»«mrow»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mo»§#160;«/mo»«mi»sin«/mi»«mo»§#160;«/mo»«mi»x«/mi»«/mrow»«mi»x«/mi»«/mfrac»«/math»” />z góry bardzo dziękuje!5f/e2/3224c3d50cd18d14b127ab76e98c.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe end cell equals cell fraction numerator open parentheses ln x close parentheses apostrophe times square root of x minus ln x times open parentheses square root of x close parentheses apostrophe over denominator open parentheses square root of x close parentheses squared end fraction equals fraction numerator begin display style 1 over x end style times square root of x minus ln x times begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style over denominator x end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator square root of x over denominator x end fraction minus fraction numerator ln x over denominator 2 square root of x end fraction end style over denominator x end fraction times fraction numerator 2 square root of x over denominator 2 square root of x end fraction equals fraction numerator begin display style fraction numerator 2 x over denominator x end fraction minus ln x end style over denominator 2 x square root of x end fraction end cell row blank equals cell fraction numerator begin display style 2 minus ln x end style over denominator 2 x square root of x end fraction end cell end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«mo»`«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mfenced»«mrow»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»`«/mo»«mo»§#183;«/mo»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«msqrt»«mi»x«/mi»«/msqrt»«/mfenced»«mo»`«/mo»«/mrow»«msup»«mfenced»«msqrt»«mi»x«/mi»«/msqrt»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»§#183;«/mo»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«/mrow»«mi»x«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mfrac»«msqrt»«mi»x«/mi»«/msqrt»«mi»x«/mi»«/mfrac»«mo»-«/mo»«mfrac»«mrow»«mi»ln«/mi»«mi»x«/mi»«/mrow»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«mi»x«/mi»«/mfrac»«mo»§#183;«/mo»«mfrac»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mstyle displaystyle=¨true¨»«mfrac»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«mi»x«/mi»«/mfrac»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” />

    I druga pochodna

    table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe apostrophe end cell equals cell open parentheses fraction numerator begin display style 2 minus ln x end style over denominator 2 x square root of x end fraction close parentheses to the power of apostrophe equals fraction numerator begin display style open parentheses 2 minus ln x close parentheses apostrophe times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x square root of x close parentheses apostrophe end style over denominator open parentheses 2 x square root of x close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x to the power of bevelled 3 over 2 end exponent close parentheses apostrophe end style over denominator open parentheses 2 x to the power of begin display style bevelled 3 over 2 end style end exponent close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times 2 times 3 over 2 x to the power of bevelled 1 half end exponent end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus open parentheses 2 minus ln x close parentheses times 3 square root of x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus 6 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 8 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style square root of x open parentheses 3 minus 8 ln x close parentheses end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 8 ln x end style over denominator 4 x squared square root of x end fraction end cell end table2a/00/aff50b4236992db275aec9e9aa54.png” alt=”equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times open parentheses negative begin display style 1 over x squared end style plus begin display style 1 over x end style close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«mo»§#183;«/mo»«mfenced»«mrow»«mo»-«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mstyle»«mo»+«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»-«/mo»«mfenced»«mrow»«mn»12«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§#183;«/mo»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«msup»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times begin display style fraction numerator negative 1 plus x over denominator x squared end fraction end style times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals

    equals fraction numerator open parentheses x minus 1 close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus 2 x squared times open parentheses 12 x cubed plus 3 x squared close parentheses times open parentheses begin display style 1 over x end style plus ln x close parentheses over denominator 2 x squared times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

    equals fraction numerator 3 x to the power of 5 plus x to the power of 4 plus x minus 3 x to the power of 4 minus x cubed minus 1 minus 24 x to the power of 4 minus 6 x cubed minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 open parentheses x times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

    equals fraction numerator 3 x to the power of 5 minus 26 x to the power of 4 minus 7 x cubed plus x minus 1 minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 times open parentheses 3 x to the power of 5 plus x to the power of 4 plus x close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction

    b0/89/71867873aebda1ce5319045f8f36.png” alt=”left right double arrow” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8660;«/mo»«/math»” />: taka obustronna strzałka jest również spójnikiem logicznym i oznacza tyle, co “wtedy i tylko wtedy”. Z lewej strony wynika prawa, a z prawej lewa. Przykład: x equals 2 space logical or space x equals negative 2 space left right double arrow space x squared equals 4 

    logical or: alternatywa – kolejny spójnik logiczny, odpowiednik słowa “lub”

    logical and: koniunkcja – spójnik logiczny, odpowiednik słowa “i”

    tilde: negacja – spójnik logiczny oznaczający zaprzeczenie

    for all: kwantyfikator duży, ogólny oznaczający “dla każdego … zachodzi …”

    there exists: kwantyfikator mały, szczegółowy oznaczający “istnieje …, takie że …”

    Więcej informacji o spójnikach logicznych można znaleźć w lekcji:
    https://online.etrapez.pl/lesson/lekcja-2-tabele-i-spojniki-logiczne-przypisywanie-wartosci-zdaniom-zlozonym/

     

    Więcej informacji o kwantyfikatorach można znaleźć w lekcji:
    https://online.etrapez.pl/wybor-kursu/matematyka-dyskretna/lekcja-29-kwantyfikatory/

    1. Tutaj jest do policzenia pochodna funkcji złożonej, czyli argumentem nie jest sam „x” tylko coś więcej, nie ma po prostu e to the power of x tylko e to the power of c o ś end exponent
      Postępujemy jak zawsze w takich przypadkach, czyli: pochodna tego co „na zewnątrz” pomnożyć razy pochodna funkcji wewnętrznej (coś więcej niż sam „x”), czyli jakby open parentheses e to the power of increment close parentheses apostrophe equals e to the power of increment times increment apostrophe .

      Stąd: open parentheses e to the power of 8 x end exponent close parentheses apostrophe equals e to the power of 8 x end exponent times open parentheses 8 x close parentheses apostrophe equals e to the power of 8 x end exponent times 8 times 1 equals 8 e to the power of 8 x end exponent 

      16/ff/8470c5a592a295d322b9588044e0.png” alt=”open parentheses a to the power of x close parentheses apostrophe equals a to the power of x times ln a” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«msup»«mi»a«/mi»«mi»x«/mi»«/msup»«/mfenced»«mo»`«/mo»«mo»=«/mo»«msup»«mi»a«/mi»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mi»a«/mi»«/math»” /> .56/59/da565d75ad307a420ee679d5b107.png” alt=”a r c sin x plus a r c cos x equals C equals a r c sin 0 plus a r c cos 0 equals 0 plus pi over 2 equals pi over 2″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»=«/mo»«mi»C«/mi»«mo»=«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mn»0«/mn»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mn»0«/mn»«mo»=«/mo»«mn»0«/mn»«mo»+«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«/math»” />

      Wtedy funkcja

      y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r c sin x plus a r c cos x close vertical bar end root equals fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 5,

      i jej pochodna

      (wg wzoru dla funkcji złożonej:  open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 triangle to the power of 4 times open parentheses triangle close parentheses apostrophe   )

      wynosi:

      y apostrophe equals fifth root of pi over 2 end root times 5 times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses sin x plus cos x close parentheses apostrophe equals

      fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses

       

      b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

       

      02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  37. michalaczek pisze:

    square root of blank end root da wpisać się do kalkulatora?d8/b5/7af3651ec6f62fa9379c5edd24aa.png” alt=”bevelled fraction numerator a r c space sin space x over denominator x end fraction” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac bevelled=¨true¨»«mrow»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mo»§#160;«/mo»«mi»sin«/mi»«mo»§#160;«/mo»«mi»x«/mi»«/mrow»«mi»x«/mi»«/mfrac»«/math»” />z góry bardzo dziękuje!5f/e2/3224c3d50cd18d14b127ab76e98c.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe end cell equals cell fraction numerator open parentheses ln x close parentheses apostrophe times square root of x minus ln x times open parentheses square root of x close parentheses apostrophe over denominator open parentheses square root of x close parentheses squared end fraction equals fraction numerator begin display style 1 over x end style times square root of x minus ln x times begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style over denominator x end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator square root of x over denominator x end fraction minus fraction numerator ln x over denominator 2 square root of x end fraction end style over denominator x end fraction times fraction numerator 2 square root of x over denominator 2 square root of x end fraction equals fraction numerator begin display style fraction numerator 2 x over denominator x end fraction minus ln x end style over denominator 2 x square root of x end fraction end cell row blank equals cell fraction numerator begin display style 2 minus ln x end style over denominator 2 x square root of x end fraction end cell end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«mo»`«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mfenced»«mrow»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»`«/mo»«mo»§#183;«/mo»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«msqrt»«mi»x«/mi»«/msqrt»«/mfenced»«mo»`«/mo»«/mrow»«msup»«mfenced»«msqrt»«mi»x«/mi»«/msqrt»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»§#183;«/mo»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«/mrow»«mi»x«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mfrac»«msqrt»«mi»x«/mi»«/msqrt»«mi»x«/mi»«/mfrac»«mo»-«/mo»«mfrac»«mrow»«mi»ln«/mi»«mi»x«/mi»«/mrow»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«mi»x«/mi»«/mfrac»«mo»§#183;«/mo»«mfrac»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mstyle displaystyle=¨true¨»«mfrac»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«mi»x«/mi»«/mfrac»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” />

    I druga pochodna

    table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe apostrophe end cell equals cell open parentheses fraction numerator begin display style 2 minus ln x end style over denominator 2 x square root of x end fraction close parentheses to the power of apostrophe equals fraction numerator begin display style open parentheses 2 minus ln x close parentheses apostrophe times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x square root of x close parentheses apostrophe end style over denominator open parentheses 2 x square root of x close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x to the power of bevelled 3 over 2 end exponent close parentheses apostrophe end style over denominator open parentheses 2 x to the power of begin display style bevelled 3 over 2 end style end exponent close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times 2 times 3 over 2 x to the power of bevelled 1 half end exponent end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus open parentheses 2 minus ln x close parentheses times 3 square root of x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus 6 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 8 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style square root of x open parentheses 3 minus 8 ln x close parentheses end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 8 ln x end style over denominator 4 x squared square root of x end fraction end cell end table2a/00/aff50b4236992db275aec9e9aa54.png” alt=”equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times open parentheses negative begin display style 1 over x squared end style plus begin display style 1 over x end style close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«mo»§#183;«/mo»«mfenced»«mrow»«mo»-«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mstyle»«mo»+«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»-«/mo»«mfenced»«mrow»«mn»12«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§#183;«/mo»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«msup»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times begin display style fraction numerator negative 1 plus x over denominator x squared end fraction end style times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals

    equals fraction numerator open parentheses x minus 1 close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus 2 x squared times open parentheses 12 x cubed plus 3 x squared close parentheses times open parentheses begin display style 1 over x end style plus ln x close parentheses over denominator 2 x squared times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

    equals fraction numerator 3 x to the power of 5 plus x to the power of 4 plus x minus 3 x to the power of 4 minus x cubed minus 1 minus 24 x to the power of 4 minus 6 x cubed minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 open parentheses x times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

    equals fraction numerator 3 x to the power of 5 minus 26 x to the power of 4 minus 7 x cubed plus x minus 1 minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 times open parentheses 3 x to the power of 5 plus x to the power of 4 plus x close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction

    b0/89/71867873aebda1ce5319045f8f36.png” alt=”left right double arrow” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8660;«/mo»«/math»” />: taka obustronna strzałka jest również spójnikiem logicznym i oznacza tyle, co “wtedy i tylko wtedy”. Z lewej strony wynika prawa, a z prawej lewa. Przykład: x equals 2 space logical or space x equals negative 2 space left right double arrow space x squared equals 4 

    logical or: alternatywa – kolejny spójnik logiczny, odpowiednik słowa “lub”

    logical and: koniunkcja – spójnik logiczny, odpowiednik słowa “i”

    tilde: negacja – spójnik logiczny oznaczający zaprzeczenie

    for all: kwantyfikator duży, ogólny oznaczający “dla każdego … zachodzi …”

    there exists: kwantyfikator mały, szczegółowy oznaczający “istnieje …, takie że …”

    Więcej informacji o spójnikach logicznych można znaleźć w lekcji:
    https://online.etrapez.pl/lesson/lekcja-2-tabele-i-spojniki-logiczne-przypisywanie-wartosci-zdaniom-zlozonym/

     

    Więcej informacji o kwantyfikatorach można znaleźć w lekcji:
    https://online.etrapez.pl/wybor-kursu/matematyka-dyskretna/lekcja-29-kwantyfikatory/

    1. Tak, “pierwiastek” można wpisać na dwa sposoby

      1) wpisując: sqrt(…)  , np sqrt(2x) oznacza square root of 2 x end root

      2) wpisując potęgę ułamkową , tzn. (…)^(1/2)  , np (x)^(1/2) oznacza square root of x

      6a/7e/3409ca399bcead514947023ac070.png” alt=”5 x to the power of 4 minus x plus 9″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»+«/mo»«mn»9«/mn»«/math»” />)be/e0/30079ee2ce183436018ad1559d9d.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe apostrophe end cell equals cell open parentheses fraction numerator begin display style 2 minus ln x end style over denominator 2 x square root of x end fraction close parentheses to the power of apostrophe equals fraction numerator begin display style open parentheses 2 minus ln x close parentheses apostrophe times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x square root of x close parentheses apostrophe end style over denominator open parentheses 2 x square root of x close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x to the power of bevelled 3 over 2 end exponent close parentheses apostrophe end style over denominator open parentheses 2 x to the power of begin display style bevelled 3 over 2 end style end exponent close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times 2 times 3 over 2 x to the power of bevelled 1 half end exponent end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus open parentheses 2 minus ln x close parentheses times 3 square root of x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus 6 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 8 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style square root of x open parentheses 3 minus 8 ln x close parentheses end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 8 ln x end style over denominator 4 x squared square root of x end fraction end cell end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«mo»`«/mo»«mo»`«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mfrac»«mstyle displaystyle=¨true¨»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mfenced»«mo»`«/mo»«/msup»«mo»=«/mo»«mfrac»«mstyle displaystyle=¨true¨»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»`«/mo»«mo»§#183;«/mo»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfenced»«mo»`«/mo»«/mstyle»«msup»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mfenced»«mrow»«mo»-«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mfrac bevelled=¨true¨»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mrow»«/mfenced»«mo»`«/mo»«/mstyle»«msup»«mfenced»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mstyle displaystyle=¨true¨»«mfrac bevelled=¨true¨»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/mstyle»«/msup»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mfenced»«mrow»«mo»-«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»2«/mn»«mo»§#183;«/mo»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«msup»«mi»x«/mi»«mfrac bevelled=¨true¨»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mo»-«/mo»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#183;«/mo»«mn»3«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mo»-«/mo»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mn»6«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»+«/mo»«mn»3«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mo»-«/mo»«mn»8«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mo»+«/mo»«mn»3«/mn»«msqrt»«mi»x«/mi»«/msqrt»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«msqrt»«mi»x«/mi»«/msqrt»«mfenced»«mrow»«mn»3«/mn»«mo»-«/mo»«mn»8«/mn»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mn»3«/mn»«mo»-«/mo»«mn»8«/mn»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” />2a/00/aff50b4236992db275aec9e9aa54.png” alt=”equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times open parentheses negative begin display style 1 over x squared end style plus begin display style 1 over x end style close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«mo»§#183;«/mo»«mfenced»«mrow»«mo»-«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mstyle»«mo»+«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»-«/mo»«mfenced»«mrow»«mn»12«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§#183;«/mo»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«msup»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times begin display style fraction numerator negative 1 plus x over denominator x squared end fraction end style times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals

      equals fraction numerator open parentheses x minus 1 close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus 2 x squared times open parentheses 12 x cubed plus 3 x squared close parentheses times open parentheses begin display style 1 over x end style plus ln x close parentheses over denominator 2 x squared times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

      equals fraction numerator 3 x to the power of 5 plus x to the power of 4 plus x minus 3 x to the power of 4 minus x cubed minus 1 minus 24 x to the power of 4 minus 6 x cubed minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 open parentheses x times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

      equals fraction numerator 3 x to the power of 5 minus 26 x to the power of 4 minus 7 x cubed plus x minus 1 minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 times open parentheses 3 x to the power of 5 plus x to the power of 4 plus x close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction

      b0/89/71867873aebda1ce5319045f8f36.png” alt=”left right double arrow” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8660;«/mo»«/math»” />: taka obustronna strzałka jest również spójnikiem logicznym i oznacza tyle, co “wtedy i tylko wtedy”. Z lewej strony wynika prawa, a z prawej lewa. Przykład: x equals 2 space logical or space x equals negative 2 space left right double arrow space x squared equals 4 

      logical or: alternatywa – kolejny spójnik logiczny, odpowiednik słowa “lub”

      logical and: koniunkcja – spójnik logiczny, odpowiednik słowa “i”

      tilde: negacja – spójnik logiczny oznaczający zaprzeczenie

      for all: kwantyfikator duży, ogólny oznaczający “dla każdego … zachodzi …”

      there exists: kwantyfikator mały, szczegółowy oznaczający “istnieje …, takie że …”

      Więcej informacji o spójnikach logicznych można znaleźć w lekcji:
      https://online.etrapez.pl/lesson/lekcja-2-tabele-i-spojniki-logiczne-przypisywanie-wartosci-zdaniom-zlozonym/

       

      Więcej informacji o kwantyfikatorach można znaleźć w lekcji:
      https://online.etrapez.pl/wybor-kursu/matematyka-dyskretna/lekcja-29-kwantyfikatory/

  38. michalaczek pisze:

    jak rozwiazac:1) f(x)=bevelled fraction numerator a r c space cos space x over denominator x end fraction2) f(x)=bevelled fraction numerator a r c space sin space x over denominator x end fractionz góry bardzo dziękuje!5f/e2/3224c3d50cd18d14b127ab76e98c.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe end cell equals cell fraction numerator open parentheses ln x close parentheses apostrophe times square root of x minus ln x times open parentheses square root of x close parentheses apostrophe over denominator open parentheses square root of x close parentheses squared end fraction equals fraction numerator begin display style 1 over x end style times square root of x minus ln x times begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style over denominator x end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator square root of x over denominator x end fraction minus fraction numerator ln x over denominator 2 square root of x end fraction end style over denominator x end fraction times fraction numerator 2 square root of x over denominator 2 square root of x end fraction equals fraction numerator begin display style fraction numerator 2 x over denominator x end fraction minus ln x end style over denominator 2 x square root of x end fraction end cell row blank equals cell fraction numerator begin display style 2 minus ln x end style over denominator 2 x square root of x end fraction end cell end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«mo»`«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mfenced»«mrow»«mi»ln«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»`«/mo»«mo»§#183;«/mo»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mfenced»«msqrt»«mi»x«/mi»«/msqrt»«/mfenced»«mo»`«/mo»«/mrow»«msup»«mfenced»«msqrt»«mi»x«/mi»«/msqrt»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»§#183;«/mo»«msqrt»«mi»x«/mi»«/msqrt»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«mo»§#183;«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«/mrow»«mi»x«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mfrac»«msqrt»«mi»x«/mi»«/msqrt»«mi»x«/mi»«/mfrac»«mo»-«/mo»«mfrac»«mrow»«mi»ln«/mi»«mi»x«/mi»«/mrow»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«mi»x«/mi»«/mfrac»«mo»§#183;«/mo»«mfrac»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«mrow»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mstyle displaystyle=¨true¨»«mfrac»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«mi»x«/mi»«/mfrac»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mstyle displaystyle=¨true¨»«mn»2«/mn»«mo»-«/mo»«mi»ln«/mi»«mi»x«/mi»«/mstyle»«mrow»«mn»2«/mn»«mi»x«/mi»«msqrt»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” />

    I druga pochodna

    table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe apostrophe end cell equals cell open parentheses fraction numerator begin display style 2 minus ln x end style over denominator 2 x square root of x end fraction close parentheses to the power of apostrophe equals fraction numerator begin display style open parentheses 2 minus ln x close parentheses apostrophe times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x square root of x close parentheses apostrophe end style over denominator open parentheses 2 x square root of x close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times open parentheses 2 x to the power of bevelled 3 over 2 end exponent close parentheses apostrophe end style over denominator open parentheses 2 x to the power of begin display style bevelled 3 over 2 end style end exponent close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style open parentheses negative 1 over x close parentheses times 2 x square root of x minus open parentheses 2 minus ln x close parentheses times 2 times 3 over 2 x to the power of bevelled 1 half end exponent end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus open parentheses 2 minus ln x close parentheses times 3 square root of x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 2 square root of x minus 6 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style negative 8 square root of x plus 3 square root of x ln x end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style square root of x open parentheses 3 minus 8 ln x close parentheses end style over denominator 4 x cubed end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 8 ln x end style over denominator 4 x squared square root of x end fraction end cell end table2a/00/aff50b4236992db275aec9e9aa54.png” alt=”equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times open parentheses negative begin display style 1 over x squared end style plus begin display style 1 over x end style close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«mo»§#183;«/mo»«mfenced»«mrow»«mo»-«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mstyle»«mo»+«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»-«/mo»«mfenced»«mrow»«mn»12«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§#183;«/mo»«msqrt»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«mo»+«/mo»«mi»ln«/mi»«mi»x«/mi»«/msqrt»«/mrow»«msup»«mfenced»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

    equals fraction numerator begin display style fraction numerator 1 over denominator 2 square root of begin display style 1 over x end style plus ln x end root end fraction end style times begin display style fraction numerator negative 1 plus x over denominator x squared end fraction end style times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus open parentheses 12 x cubed plus 3 x squared close parentheses times square root of begin display style 1 over x end style plus ln x end root over denominator open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared end fraction equals

    equals fraction numerator open parentheses x minus 1 close parentheses times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses minus 2 x squared times open parentheses 12 x cubed plus 3 x squared close parentheses times open parentheses begin display style 1 over x end style plus ln x close parentheses over denominator 2 x squared times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

    equals fraction numerator 3 x to the power of 5 plus x to the power of 4 plus x minus 3 x to the power of 4 minus x cubed minus 1 minus 24 x to the power of 4 minus 6 x cubed minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 open parentheses x times open parentheses 3 x to the power of 4 plus x cubed plus 1 close parentheses close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction equals

    equals fraction numerator 3 x to the power of 5 minus 26 x to the power of 4 minus 7 x cubed plus x minus 1 minus open parentheses 24 x to the power of 5 plus 6 x to the power of 4 close parentheses times ln x over denominator 2 times open parentheses 3 x to the power of 5 plus x to the power of 4 plus x close parentheses squared times square root of begin display style 1 over x end style plus ln x end root end fraction

    b0/89/71867873aebda1ce5319045f8f36.png” alt=”left right double arrow” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8660;«/mo»«/math»” />: taka obustronna strzałka jest również spójnikiem logicznym i oznacza tyle, co “wtedy i tylko wtedy”. Z lewej strony wynika prawa, a z prawej lewa. Przykład: x equals 2 space logical or space x equals negative 2 space left right double arrow space x squared equals 4 

    logical or: alternatywa – kolejny spójnik logiczny, odpowiednik słowa “lub”

    logical and: koniunkcja – spójnik logiczny, odpowiednik słowa “i”

    tilde: negacja – spójnik logiczny oznaczający zaprzeczenie

    for all: kwantyfikator duży, ogólny oznaczający “dla każdego … zachodzi …”

    there exists: kwantyfikator mały, szczegółowy oznaczający “istnieje …, takie że …”

    Więcej informacji o spójnikach logicznych można znaleźć w lekcji:
    https://online.etrapez.pl/lesson/lekcja-2-tabele-i-spojniki-logiczne-przypisywanie-wartosci-zdaniom-zlozonym/

     

    Więcej informacji o kwantyfikatorach można znaleźć w lekcji:
    https://online.etrapez.pl/wybor-kursu/matematyka-dyskretna/lekcja-29-kwantyfikatory/

    1. 1. f open parentheses x close parentheses equals fraction numerator a r c cos x over denominator x end fraction

      Skorzystam ze wzoru:

      open parentheses u over v close parentheses apostrophe equals fraction numerator u apostrophe v minus u v apostrophe over denominator v squared end fraction

      f apostrophe open parentheses x close parentheses equals open parentheses fraction numerator a r c cos x over denominator x end fraction close parentheses apostrophe equals fraction numerator open parentheses a r c cos x close parentheses apostrophe times x minus a r c cos x times open parentheses x close parentheses apostrophe over denominator x squared end fraction equals

      fraction numerator negative begin display style fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction end style times x minus a r c cos x times 1 over denominator x squared end fraction equals negative fraction numerator begin display style fraction numerator x over denominator square root of 1 minus x squared end root end fraction end style plus a r c cos x over denominator x squared end fraction equals

      negative fraction numerator x plus square root of 1 minus x squared end root times a r c cos x over denominator x squared times square root of 1 minus x squared end root end fraction

      48/d2/4b6181539cac56b2709119d6442e.png” alt=”D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»D«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»§#8477;«/mi»«mo»«/mo»«mo»{«/mo»«mo»-«/mo»«mn»1«/mn»«mo»,«/mo»«mn»2«/mn»«mo»}«/mo»«/math»” />.

      Przechodzimy do wyznaczania monotoniczności funkcji f. W tym celu obliczymy jej pochodną i sprawdzimy, kiedy jest dodatnia, a kiedy ujemna.

      f apostrophe left parenthesis x right parenthesis equals fraction numerator open parentheses x cubed close parentheses apostrophe times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses x squared minus x minus 2 close parentheses apostrophe over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals
      equals fraction numerator 3 x squared times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses 2 x minus 1 close parentheses over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals fraction numerator 3 x to the power of 4 minus 3 x cubed minus 6 x squared minus 2 x to the power of 4 plus x cubed over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals
      equals fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction

      Zbadamy teraz, kiedy pochodna przyjmuje wartości większe lub równe 0, a kiedy mniejsze lub równe 0.

      fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  39. Karolina pisze:

    czy mógłby ktoś mi pomóc z rozwiązaniem pochodnej: (x^2)/(2-x) ?
    Kalkulator wylicza to jako: [-(x-4)x]/[(x-2)^2]
    Ja wyliczam już czwarty raz i za każdym wychodzi mi taki sam wynik [(4-x)x]/[(2-x)^2], niestety inny niż kalkulatora 🙁
    proszę o pomoc!

    1. Joanna Grochowska pisze:

      Oba wyniki są poprawne i oba są identyczne 🙂

      Po prostu ten z kalkulatora wyliczony “wyciągnął” jeszcze minusy z każdego z wyrażeń.

      Przekształcę więc je tak, że na górze wciągnę go z powrotem, a na dole jakby go wyciągnę jeszcze raz (bo podniesiony do kwadratu się zredukował). Proszę popatrzeć:

      \displaystyle \frac{{-(x-4)x}}{{{{{(x-2)}}^{2}}}}=\frac{{(-x+4)x}}{{{{{\left[ {-(-x+2)} \right]}}^{2}}}}=\frac{{(4-x)x}}{{{{{(-1)}}^{2}}{{{(2-x)}}^{2}}}}=\frac{{(4-x)x}}{{{{{(2-x)}}^{2}}}}

      No i wyszedł Pani wynik 🙂

    2. Karolina pisze:

      jeju, rzeczywiście, ale głupi błąd! 😛
      Bardzo dziękuje, juz rozumiem 😉

  40. Julia pisze:

    Witam, nie rozumiem dlaczego pochodna ln2x^2 to y\'(x) = (2 log(2 x))\/x

  41. Matematyk pisze:

    Bardzo pomocny kalkulator pochodnych funkcji, przydatny szczególnie do sprawdzania wyników.

  42. Paulina pisze:

    Witam 🙂 Dlaczego pochodna z e^(2^x)=2^x*e^(2^x)*log2?

    1. Tutaj jest do policzenia pochodna funkcji złozonej, czyli argumentem nie jest sam „x” tylko coś więcej, nie ma po prostu e to the power of x tylko e to the power of c o ś end exponent.
      Postępujemy jak zawsze w takich przypadkach, czyli: pochodna tego co „na zewnątrz” pomnożyć razy pochodna funkcji wewnętrznej (coś więcej niż sam „x”), czyli jakby open parentheses e to the power of increment close parentheses apostrophe equals e to the power of increment times increment apostrophe

      Stąd: open parentheses e to the power of 2 to the power of x end exponent close parentheses apostrophe equals e to the power of 2 to the power of x end exponent times open parentheses 2 to the power of x close parentheses apostrophe equals e to the power of 2 to the power of x end exponent times 2 to the power of x times ln 2 , gdyż wprost z wzorku  open parentheses a to the power of x close parentheses apostrophe equals a to the power of x times ln a .

      56/59/da565d75ad307a420ee679d5b107.png” alt=”a r c sin x plus a r c cos x equals C equals a r c sin 0 plus a r c cos 0 equals 0 plus pi over 2 equals pi over 2″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»=«/mo»«mi»C«/mi»«mo»=«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mn»0«/mn»«mo»+«/mo»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»cos«/mi»«mn»0«/mn»«mo»=«/mo»«mn»0«/mn»«mo»+«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mi»§#960;«/mi»«mn»2«/mn»«/mfrac»«/math»” />

      Wtedy funkcja

      y equals open parentheses sin x plus cos x close parentheses to the power of 5 times fifth root of open vertical bar a r c sin x plus a r c cos x close vertical bar end root equals fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 5,

      i jej pochodna

      (wg wzoru dla funkcji złożonej:  open parentheses triangle to the power of 5 close parentheses apostrophe equals 5 triangle to the power of 4 times open parentheses triangle close parentheses apostrophe   )

      wynosi:

      y apostrophe equals fifth root of pi over 2 end root times 5 times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses sin x plus cos x close parentheses apostrophe equals

      fifth root of pi over 2 end root times open parentheses sin x plus cos x close parentheses to the power of 4 times open parentheses cos x minus sin x close parentheses

       

      b6/23/d4828ea2d1df0b14e59024956237.png” alt=”4 over 3 x to the power of negative 2 over 3 end exponent times fraction numerator 5 x squared plus 5 plus 2 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 minus 30 x squared minus 12 x squared times 4 to the power of 3 to the power of x end exponent over denominator open parentheses x squared plus 1 close parentheses squared end fraction equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/msup»«mo»§#183;«/mo»«mfrac»«mrow»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«mo»§#183;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»ln«/mi»«mn»4«/mn»«mo»§#183;«/mo»«mi»ln«/mi»«mn»3«/mn»«mo»-«/mo»«mn»30«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»12«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#183;«/mo»«msup»«mn»4«/mn»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«/math»” />

      4 over 3 times fraction numerator negative 25 x squared plus 5 minus 10 x squared times 4 to the power of 3 to the power of x end exponent plus 2 times 4 to the power of 3 to the power of x end exponent plus 6 x cubed times 4 to the power of 3 to the power of x end exponent times 3 to the power of x times ln 4 times ln 3 plus 6 x times 4 to the power of 3 to the power of x end exponent times ln 4 times ln 3 over denominator x to the power of begin display style 2 over 3 end style end exponent times open parentheses x squared plus 1 close parentheses squared end fraction

       

      02/6f/450ef1d93789d392f640d05061c5.png” alt=”fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«mo»§#8805;«/mo»«mn»0«/mn»«/math»” />

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  43. studentekonomi pisze:

    Witam. Mam problem z zadaniem: f(x1,x2)=1/2ln(5×1^2-2×2). Jak mogę narysować krzywe w punktach 0, 1 i 2? Wytyczenie pochodnej i całki również by się przydało…

  44. Iulia pisze:

    Dzień dobry, bardzo prosiłabym o pomóc z przykładem [((arctgX^2)^3)/((e^3)*x+3^x)]^(arctg(x^4-ln(2x^8+1) Czyli iloraz w tym kwadratowym nawiasie podnosimy do potęgi i z tego wszystkiego policzyć pochodną…wychodzą mi kosmiczne rozwiazania…Z góry dziękuję.

  45. Kasia pisze:

    Panie Krystianie,
    może jest mi Pan w stanie wytłumaczyć dlaczego pochodna z -arctg|x| ma pochodną -x/(|x^3|+|x|), a nie po prostu -1/(1+x^2)?

    Byłabym bardzo wdzięczna za pomoc 🙂

    1. Joanna Grochowska pisze:

      Pani Kasiu, gdyby do policzenia byłaby pochodna po prostu z displaystyle -arctgxto byłaby równa rzeczywiście displaystyle -frac{1}{{1+{{x}^{2}}}}

      Jednak tutaj do policzenia jest pochodna displaystyle -arctgleft| x right|, czyli argumentem nie jest sam “x” tylko coś więcej – moduł z “x”.

      Postępujemy jak zawsze w takich przypadkach, czyli: pochodna tego co “na zewnątrz” pomnożyć razy pochodna funkcji wewnętrznej (coś więcej niż sam “x”), czyli jakby displaystyle left( {-arctgDelta } right)'cdot Delta '

      Pytanie, ile wynosi pochodna modułu z x ?

      Rozpisując moduł, wiem, że:
      open vertical bar x close vertical bar equals open curly brackets table attributes columnalign left end attributes row cell x comma space space space space space space space space x greater or equal than 0 end cell row cell negative x comma space space space space space x less than 0 end cell end table close

      Czyli odpowiednio pochodna byłby równa 1 lub -1.. Jednak potrzebuję pochodnej w ogólnym przypadku (nie na przedziałach).

      Dlatego uznaje się, że pochodna modułu to (warto zapamiętać ten wzór):

      displaystyle left( {left| x right|} right)'=frac{x}{{left| x right|}}

      Można sobie rozpisać na odpowiednich przedziałach i faktycznie wyjdzie 1 lub -1 😉

      Mając wszystko, liczę:

      displaystyle left( {-arctgleft| x right|} right)'=-frac{1}{{1+{{{left| x right|}}^{2}}}}cdot left( {left| x right|} right)'=-frac{1}{{1+{{{left| x right|}}^{2}}}}cdot frac{x}{{left| x right|}}=-frac{x}{{left| x right|+{{{left| x right|}}^{3}}}}20/f8/ac8d5e8f241d98bb3025a456efd5.png” alt=”f colon space X rightwards arrow Y” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»:«/mo»«mo»§#160;«/mo»«mi»X«/mi»«mo»§#8594;«/mo»«mi»Y«/mi»«/math»” />” oznacza, że funkcja f ma dziedzinę w zbiorze X i zbiór wartości w zbiorze Y

      rightwards double arrow: taka strzałka jest jednym ze spójników logicznych i oznacza wynikanie. Przykład:  Zdanie “Jeśli x equals 2, to  x squared equals 4 .” można zapisać następująco “x equals 2 space rightwards double arrow space x squared equals 4“. Jest to wynikanie w jedną stronę, zgodną ze zwrotem strzałki.

      left right double arrow: taka obustronna strzałka jest również spójnikiem logicznym i oznacza tyle, co “wtedy i tylko wtedy”. Z lewej strony wynika prawa, a z prawej lewa. Przykład: x equals 2 space logical or space x equals negative 2 space left right double arrow space x squared equals 4 

      logical or: alternatywa – kolejny spójnik logiczny, odpowiednik słowa “lub”

      logical and: koniunkcja – spójnik logiczny, odpowiednik słowa “i”

      tilde: negacja – spójnik logiczny oznaczający zaprzeczenie

      for all: kwantyfikator duży, ogólny oznaczający “dla każdego … zachodzi …”

      there exists: kwantyfikator mały, szczegółowy oznaczający “istnieje …, takie że …”

      Więcej informacji o spójnikach logicznych można znaleźć w lekcji:
      https://online.etrapez.pl/lesson/lekcja-2-tabele-i-spojniki-logiczne-przypisywanie-wartosci-zdaniom-zlozonym/

       

      Więcej informacji o kwantyfikatorach można znaleźć w lekcji:
      https://online.etrapez.pl/wybor-kursu/matematyka-dyskretna/lekcja-29-kwantyfikatory/

  46. Kamil pisze:

    Mam wielką prośbę. Nie moge poradzić sobie z monotonicznością tej funkcji x^3/(x^2+-x-2) będę ogromnie wdzięczny za odpowiedz. Pozdrawiam 🙂

    1. Kamil pisze:

      x^3/(x^2-x-2) wyzej jest mały bląd

    2. Anna Zalewska pisze:

      Dana jest funkcja f left parenthesis x right parenthesis equals fraction numerator x cubed over denominator x squared minus x minus 2 end fraction.

      Zaczynamy od wyznaczenia dziedziny funkcji.

      x squared minus x minus 2 not equal to 0
      capital delta equals left parenthesis negative 1 right parenthesis squared minus 4 times 1 times left parenthesis negative 2 right parenthesis equals 9
      x subscript 1 equals fraction numerator 1 minus square root of 9 over denominator 2 end fraction equals fraction numerator 1 minus 3 over denominator 2 end fraction equals fraction numerator negative 2 over denominator 2 end fraction equals negative 1
      space x subscript 2 equals fraction numerator 1 plus square root of 9 over denominator 2 end fraction equals fraction numerator 1 plus 3 over denominator 2 end fraction equals 4 over 2 equals 2

      Zatem D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

      Przechodzimy do wyznaczania monotoniczności funkcji f. W tym celu obliczymy jej pochodną i sprawdzimy, kiedy jest dodatnia, a kiedy ujemna.

      f apostrophe left parenthesis x right parenthesis equals fraction numerator open parentheses x cubed close parentheses apostrophe times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses x squared minus x minus 2 close parentheses apostrophe over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals
      equals fraction numerator 3 x squared times open parentheses x squared minus x minus 2 close parentheses minus x cubed times open parentheses 2 x minus 1 close parentheses over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals fraction numerator 3 x to the power of 4 minus 3 x cubed minus 6 x squared minus 2 x to the power of 4 plus x cubed over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction equals
      equals fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction

      Zbadamy teraz, kiedy pochodna przyjmuje wartości większe lub równe 0, a kiedy mniejsze lub równe 0.

      fraction numerator x to the power of 4 minus 2 x cubed minus 6 x squared over denominator open parentheses x squared minus x minus 2 close parentheses squared end fraction greater or equal than 0

      x to the power of 4 minus 2 x cubed minus 6 x squared greater or equal than 0

      x squared open parentheses x squared minus 2 x minus 6 close parentheses greater or equal than 0

      capital delta subscript 1 equals left parenthesis negative 2 right parenthesis squared minus 4 times 1 times left parenthesis negative 6 right parenthesis equals 28
      x subscript 1 equals fraction numerator 2 minus square root of 28 over denominator 2 end fraction equals fraction numerator 2 minus 2 square root of 7 over denominator 2 end fraction equals 1 minus square root of 7
      x subscript 2 equals fraction numerator 2 plus square root of 28 over denominator 2 end fraction equals fraction numerator 2 plus 2 square root of 7 over denominator 2 end fraction equals 1 plus square root of 7
      wykres

      Pochodna przyjmuje wartości większe lub równe 0 dla x element of left parenthesis negative infinity comma 1 minus square root of 7 greater than oraz dla x element of less than 1 plus square root of 7 comma space plus infinity right parenthesis
      Pochodna przyjmuje wartości mniejsze lub równe 0 dla x element of less than 1 minus square root of 7 comma 1 plus square root of 7 greater than

       

      Należy pamiętać o założeniach dziedziny: D equals straight real numbers backslash left curly bracket negative 1 comma 2 right curly bracket.

       

      Zatem podana funkcja jest rosnąca w przedziałach x element of left parenthesis negative infinity comma 1 minus square root of 7 greater thanx element of less than 1 plus square root of 7 comma space plus infinity right parenthesis oraz malejąca w przedziałach x element of less than 1 minus square root of 7 comma negative 1 right parenthesisx element of open parentheses negative 1 comma 2 close parenthesesx element of left parenthesis 2 comma space 1 plus square root of 7 greater than.

  47. Klaudia pisze:

    Witam! Mam taką funkcję :
    f(x) = (2x-x^2)^(2/3). Jak obliczyć pochodną takiej funkcji?

    1. Joanna Grochowska pisze:

      By obliczyć pochodną z funkcji \displaystyle {{(2x-{{x}^{2}})}^{{\frac{2}{3}}}} stosuję wzór

      \displaystyle \left( {{{x}^{n}}} \right)'=n\cdot {{x}^{{n-1}}}, gdzie jak zauważam, mam coś więcej niż sam “x”, mam dodatkową funkcję (zwaną funkcją wewnętrzną). W taki przypadku obliczoną pochodną przemnażamy przez pochodną funkcji wewnętrznej, czyli mam jakby:

      \displaystyle \left( {{{\Delta }^{n}}} \right)'=n\cdot {{\Delta }^{{n-1}}}\cdot \Delta '

      Mam więc:
      \displaystyle \left( {{{{(2x-{{x}^{2}})}}^{{\frac{2}{3}}}}} \right)'=\frac{2}{3}{{(2x-{{x}^{2}})}^{{\frac{2}{3}-1}}}\cdot (2x-{{x}^{2}})'=\frac{2}{3}{{(2x-{{x}^{2}})}^{{-\frac{1}{3}}}}\cdot (2-2x)=\frac{{2\cdot (2-2x)}}{{3\sqrt[3]{{2x-{{x}^{2}}}}}}

  48. Karim pisze:

    Witam wszystkich. I proszę o pomoc.
    Mam problem z taką pochodną
    f(x)=[1-sin(2x)]/[2x^4+7x^2-3] Zatrzymuje się w pewnym momencie i nie wiem co dalej. Kalkulator do pochodnych stworzonego przez Pana Krystiana błędnie odczytuje ostatnia część 7x^2-3 zamiast zrobić wszystko w potędze obejmuje liczbę trzy od reszty za potęga. Proszę o pomoc

    1. Joanna Grochowska pisze:

      To nie chodzi Panu o pochodną funkcji \displaystyle \frac{{1-sin(2x)}}{{2{{x}^{4}}+7{{x}^{2}}-3}}?

      A może \displaystyle \frac{{1-sin(2x)}}{{2{{x}^{4}}+{{7}^{{{{x}^{2}}-3}}}}}, czy jeszcze inaczej? Proszę może gdzieś nawias () wstawić dodatkowo, to co ma być ujęte w potędze, bo nie do końca rozumiem o co chodzi z
      “część 7x^2-3 zamiast zrobić wszystko w potędze obejmuje liczbę trzy od reszty za potęga”.

      Pozdrawiam

  49. aga pisze:

    Witam, mam problem z pochodną: e^(3x+2)*((x^6)+4). Nie mam pojęcia jak to rozwiązać, bardzo proszę o pomoc…

    1. Joanna Grochowska pisze:

      Wykorzystuję tutaj wzór na iloczyn dwóch funkcji

      \displaystyle \left( {f\cdot g} \right)'=f'\cdot g+f\cdot g'

      Muszę również pamiętać o tym, że licząc pochodną funkcji złożonej, muszę domnożyć jeszcze razy pochodna funkcji wewnętrznej, tego “coś więcej niż sam x” . to znaczy

      \displaystyle \left( {{{e}^{\Delta }}} \right)'={{e}^{\Delta }}\cdot \Delta '

      No to rozwiązując przykład:
      \displaystyle \begin{array}{l}\left( {{{e}^{{3x+2}}}\cdot ({{x}^{6}}+4)} \right)'=\left( {{{e}^{{3x+2}}}} \right)'\cdot ({{x}^{6}}+4)+{{e}^{{3x+2}}}\cdot ({{x}^{6}}+4)'=\\{{e}^{{3x+2}}}\cdot (3x+2)'\cdot ({{x}^{6}}+4)+{{e}^{{3x+2}}}\cdot (6{{x}^{5}}+0)={{e}^{{3x+2}}}\cdot 3\cdot ({{x}^{6}}+4)+{{e}^{{3x+2}}}\cdot 6{{x}^{5}}=\\3{{e}^{{3x+2}}}\cdot \left( {{{x}^{6}}+4+2{{x}^{5}}} \right)=3{{e}^{{3x+2}}}\cdot \left( {{{x}^{6}}+2{{x}^{5}}+4} \right)\end{array}

  50. Andzia pisze:

    Pochodna z: cos^2pierwiastek z x +sin^2pierwiastek z x.

    1. Joanna Grochowska pisze:

      Czyli chodzi o pochodną funkcji \displaystyle {{\cos }^{2}}\sqrt{x}+{{\sin }^{2}}\sqrt{x}?

      No to liczę:
      \displaystyle \begin{array}{l}\left( {{{{\cos }}^{2}}\sqrt{x}+{{{\sin }}^{2}}\sqrt{x}} \right)'=2\cos \sqrt{x}\cdot \left( {\cos \sqrt{x}} \right)'+2\sin \sqrt{x}\cdot \left( {\sin \sqrt{x}} \right)'=\\2\cos \sqrt{x}\cdot (-\sin \sqrt{x})\cdot \left( {\sqrt{x}} \right)'+2\sin \sqrt{x}\cdot \cos \sqrt{x}\cdot \left( {\sqrt{x}} \right)'=\\-2\sin \sqrt{x}\cos \sqrt{x}\cdot \frac{1}{{2\sqrt{x}}}+2\sin \sqrt{x}\cdot \cos \sqrt{x}\cdot \frac{1}{{2\sqrt{x}}}=0\end{array}

  51. Misia pisze:

    Dzień dobry panie Krystianie, czy mogłabym liczyć na pomoc w policzeniu pochodnej e^-x^2
    Z góry dziękuję i pozdrawiam

    1. Joanna Grochowska pisze:

      Pochodna funkcji \displaystyle y={{e}^{-}}^{{{{x}^{2}}}}

      Jest to funkcja złożona, licząc jej pochodną, liczę pochodną funkcji “zewnętrznej”, czyli e^(coś) i muszę domnożyć jeszcze ją razy pochodna funkcji wewnętrznej, tego „coś więcej niż sam x” . to znaczy

      \displaystyle \left( {{{e}^{\Delta }}} \right)'={{e}^{\Delta }}\cdot \Delta '

      Mam:

      \displaystyle \left( {{{e}^{-}}^{{{{x}^{2}}}}} \right)'={{e}^{-}}^{{{{x}^{2}}}}\cdot \left( {-{{x}^{2}}} \right)'={{e}^{-}}^{{{{x}^{2}}}}\cdot \left( {-2x} \right)=-2x{{e}^{-}}^{{{{x}^{2}}}}

  52. ela pisze:

    Witam a jak to rozwiązać ? :/ (x+1)(x+4)

    1. Joanna Grochowska pisze:

      f(x)=(x+1)(x+4)

      Pochodną tego można policzyć tak na prawdę na dwa sposoby:

      I SPOSÓB – z pochodnej iloczynu \displaystyle \left( {f\cdot g} \right)'=f'\cdot g+f\cdot g'

      \displaystyle \begin{array}{l}\left( {\text{(x+1)(x+4)}} \right)\text{ }\!\!'\!\!\text{ =(x+1) }\!\!'\!\!\text{ }\cdot \text{(x+4)}+\text{(x+1)}\cdot \text{(x+4) }\!\!'\!\!\text{ =(1+0)}\cdot \text{(x+4)}+\text{(x+1)}\cdot \text{(1+0)=}\\\text{x+4+x+1=2x+5}\end{array}

      II SPOSÓB – przemnożyć przez siebie te dwa nawiasy (bez problemu mogę, gdyż w jednym jak i w drugim jest wielomian) i potem policzyć pochodną otrzymanego wielomianu korzystając z wzoru \displaystyle \left( {{{x}^{n}}} \right)'=n\cdot {{x}^{{n-1}}}

      \displaystyle \text{(x+1)(x+4)}={{x}^{2}}+4x+x+4={{x}^{2}}+5x+4

      \displaystyle \left( {{{x}^{2}}+5x+4} \right)'=\left( {{{x}^{2}}} \right)'+\left( {5x} \right)'+\left( 4 \right)'=2x+5\cdot 1+0=2x+5

  53. Lidka pisze:

    Witam Panie Krystianie. Czy w wyznaczaniu pochodnych takie cos jak: e^pi , traktujemy jako liczbę czyli wynik to zero czy w inny sposób?

    Dziękuje za odpowiedz
    Pozdrawiam

    1. Joanna Grochowska pisze:

      Tak dokładnie, traktujemy to wyrażenie jako liczbę (nie ma Pani tutaj żadnej zmiennej „x”, tylko same stałe), więc pochodna tego to zero 🙂

  54. Czy ktoś by mógł mi pomóc w rozwiązaniu tych pochodnych?

    y=e^(1/cosx)
    y=a/2(e^(x/a)+e^(-(x/a)))
    y=arcsin(e^4x )
    y=e^√(7x^2 )
    y=log_7cos√(1+x)

    1. Joanna Grochowska pisze:

      Przykład pierwszy: \displaystyle y={{e}^{{\frac{1}{{\cos x}}}}}

      Jest to funkcja złożona, liczę pochodną funkcji „zewnętrznej”, czyli e^(coś) i muszę domnożyć jeszcze ją razy pochodna funkcji wewnętrznej, tego „coś więcej niż sam x” . To znaczy

      \displaystyle \left( {{{e}^{\Delta }}} \right)'={{e}^{\Delta }}\cdot \Delta '

      Mam:
      \displaystyle \left( {{{e}^{{\frac{1}{{\cos x}}}}}} \right)'={{e}^{{\frac{1}{{\cos x}}}}}\cdot \left( {\frac{1}{{\cos x}}} \right)'

      Pochodną \displaystyle \left( {\frac{1}{{\cos x}}} \right)'można policzyć np z wzoru na iloraz dwóch funkcji
      \displaystyle \left( {\frac{f}{g}} \right)'=\frac{{f'\cdot g-f\cdot g'}}{{{{g}^{2}}}}

      \displaystyle \begin{array}{l}\left( {{{e}^{{\frac{1}{{\cos x}}}}}} \right)'={{e}^{{\frac{1}{{\cos x}}}}}\cdot \left( {\frac{1}{{\cos x}}} \right)'={{e}^{{\frac{1}{{\cos x}}}}}\cdot \frac{{\left( 1 \right)'\cdot \cos x-1\cdot (\cos x)'}}{{{{{\cos }}^{2}}x}}=\\{{e}^{{\frac{1}{{\cos x}}}}}\cdot \frac{{0\cdot \cos x-1\cdot (-\sin x)}}{{{{{\cos }}^{2}}x}}={{e}^{{\frac{1}{{\cos x}}}}}\cdot \frac{{\sin x}}{{{{{\cos }}^{2}}x}}={{e}^{{\frac{1}{{\cos x}}}}}\cdot tgx\cdot \frac{1}{{\cos x}}\end{array}

    2. Joanna Grochowska pisze:

      Przykład drugi: \displaystyle y=\frac{a}{2}({{e}^{{^{{\frac{x}{a}}}}}}+{{e}^{{-\frac{x}{a}}}})

      Jak rozumiem, liczbę “a” traktuję jako pewną stałą?

      No to liczę pochodną, stosując wzór: \displaystyle \left( {{{e}^{\Delta }}} \right)'={{e}^{\Delta }}\cdot \Delta '

      \displaystyle y'=\left( {\frac{a}{2}({{e}^{{^{{\frac{x}{a}}}}}}+{{e}^{{-\frac{x}{a}}}})} \right)'=\frac{a}{2}\left( {{{e}^{{^{{\frac{x}{a}}}}}}+{{e}^{{-\frac{x}{a}}}}} \right)'=\frac{a}{2}\left[ {\left( {{{e}^{{^{{\frac{x}{a}}}}}}} \right)'+\left( {{{e}^{{-\frac{x}{a}}}}} \right)'} \right]=
      \displaystyle \frac{a}{2}\left[ {{{e}^{{^{{\frac{x}{a}}}}}}\left( {\frac{x}{a}} \right)'+{{e}^{{^{{-\frac{x}{a}}}}}}\left( {-\frac{x}{a}} \right)'} \right]=\frac{a}{2}\left[ {{{e}^{{^{{\frac{x}{a}}}}}}\cdot \frac{1}{a}\cdot 1+{{e}^{{^{{-\frac{x}{a}}}}}}\cdot \left( {-\frac{1}{a}} \right)\cdot 1} \right]=
      \displaystyle \frac{a}{2}\cdot \frac{1}{a}\left( {{{e}^{{^{{\frac{x}{a}}}}}}-{{e}^{{^{{-\frac{x}{a}}}}}}} \right)=\frac{1}{2}\left( {{{e}^{{^{{\frac{x}{a}}}}}}-{{e}^{{^{{-\frac{x}{a}}}}}}} \right)

      Można ewentualnie trochę przekształcić wynik i otrzymać:
      \displaystyle \frac{1}{2}\left( {{{e}^{{^{{\frac{x}{a}}}}}}-{{e}^{{^{{-\frac{x}{a}}}}}}