The Derivative of the Square Root of x – How to Derive the Formula
Krystian Karczyński
Founder and General Manager of eTrapez.
Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.
He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.
Deriving Formulas for Derivatives
Derivative formulas didn’t just come from outer space, they are actually derived from the definition of a derivative:
The Derivative of the Square Root of x
Let’s derive the formula for the derivative of the square root of x: 
. We should get: 
(so say the basic derivative formulas – formula number 5).
We have 
. Let’s get to it. After substituting into the derivative formula from the definition, we will receive:
By multiplying the numerator and the denominator in the following way…
…and using the formula for perfect square trinomial in the numerator, we will eliminate the irrationality and end up with:
The x’s on top in the numerator will cancel out and we get…
…and after cancelling out 
in the numerator and denominator:
Since 
it means that:
So, we are home. The formula for the derivative has been derived.
More Examples
Feel free to play around with other formulas, I encourage you!
Derivatives and Function Analysis Course
You can also find various step-by-step calculated cases in my Derivatives and Function Analysis Course.
More General Cases
The task of deriving a formula for a derivative always comes down to calculating the appropriate limit, where ‘x’ is treated as a constant. It might be easier or harder, but you can use methods and tricks you’re already familiar with from calculating function limits.
With one caveat.
Unfortunately – the l’Hôpital’s Rule is off the table. Why? Precisely because it utilizes derivatives.
Let me remind you of your task – you need to compute the derivative of a function from the definition, without knowing the formula. And l’Hôpital’s Rule often uses derivative formulas!
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