## Formulas for Derivatives

### Topic: Theorem on the Derivative of an Inverse Function. Applying the Theorem to Derive Several Derivative Formulas.

#### Summary

In our lecture, we’re gonna dive into this cool theorem about the derivative of an inverse function. We’ll prove it and see how to use it to figure out a bunch of derivative formulas. Without this theorem, deriving these (like we did directly from definitions in the last lecture) would be, to put it mildly – a tough cookie to crack.

Before tackling the theorem, it’d be great to know what an inverse function is, why the inverse of is and why we need to limit ourselves in this case to a range of arguments like …

Using the theorem to derive formulas is a pretty common task in university-level calculus, so this lecture might just save your academic life one day.

I’m pulling the theorem and proof from Fichtenholz’s book, tweaking it here and there, fixing typos, and making a few alterations.

#### Theorem on the Derivative of an Inverse Function

*If the function * *has an inverse function* *and at point it has a finite and non-zero derivative , then at the corresponding point the derivative of the inverse function exists and its value at is .*

Confused by the string of symbols? At first, it’s very likely, but let’s sink our teeth into this theorem with a couple of simple, concrete examples.

Alright, let’s translate the excerpt into English with a casual and slightly playful tone, including the specialized mathematical terminology:

—

**Example 1**

If the functionhas an inverse function,

1. Let’s take the function

2. Its inverse function exists and is

and at the pointit has a finite and non-zero derivative ,

3. Let’s consider the point

then at the corresponding pointthe point

4. The corresponding point to

So in our example:

the derivative of the inverse functionexists

5. Indeed, the inverse function is

and its value at the pointis equal to .

6. Indeed, the value calculated in point 5.

(

**So the Theorem “works” 🙂**

Sure, here’s the translation of the specified excerpt from the math blog into English, maintaining a casual and slightly humorous tone:

**Example 2**

If the functionhas an inverse function,

1. Let’s take the exponential function

2. Its inverse function exists and is

and at the pointit has a finite and non-zero derivative ,

3. Let’s take the point

then at the corresponding pointthe point

4. The corresponding point to , which is

So:

the derivative of the inverse functionexists

5. Indeed, the inverse function is

and its value at the pointis equal to .

6. Indeed, calculated at point 5.

(

**So the Theorem “works” again 🙂**

#### Proof of the Theorem on the Derivative of the Inverse Function

We’ll prove this theorem by referring to the geometric interpretation of the derivative of a function at a point. As we remember, the value of the derivative of a function at a point is the tangent of the slope of the tangent line to the graph of the function at that point.

Graphically, it would look like this:

we defined in previous lectures as the tangent of angle

Now let’s notice something interesting: the graph of the inverse function to

equals:

So, we see that the derivative values of the function and its inverse function are the tangents of angles in the same right triangle.

And such tangents of angles in a right triangle (as we remember from high school) are related by the relation:

Thus (after dividing both sides by

From this follows the conclusion of our theorem, namely:

🙂

END OF PROOF

#### Deriving Formulas for Derivatives Using the Theorem on the Derivative of an Inverse Function

**Example 3**

*Let’s derive the formula for the derivative of the function .*

The formula we need to derive is:

Our function f(x) is the arccosx function. The inverse function to it is

According to the theorem on the derivative of an inverse function, the value of the derivative of the inverse function at point is equal to the reciprocal of the value of the derivative of the function at point

*:*

So, at any point :

After transforming:

Using the trigonometric identity, we can derive that:

Now, pay attention: is the value of the function

*at the point*

*, which is*

* *(obviously satisfying domain conditions, which I neglected), so our formula

**Example 4**

*Derive the formula for the derivative of the function .*

The formula we need to derive is:

Our function f(x) is the arctgx function. The inverse function to it is

According to the theorem on the derivative of an inverse function, the value of the derivative of the inverse function at point is equal to the reciprocal of the value of the derivative of the function at point

*:*

So, at any point :

After transforming:

Using the trigonometric identity, we can further transform it:

Now, pay attention: is the value of the function

(obviously satisfying domain conditions, which I neglected), so our formula

THE END

While writing this post, I used…

1. “Differential and Integral Calculus. Volume I.” by G.M. Fichtenholz. Published in 1966.

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