Derivative of Inverse Function

There are many electronic devices that can be used to gather information about natural phenomena. Let's think of an electrocardiogram for example, which is a device that gathers information about our heartbeats and displays it on a screen.

The screen of an electrocardiogram

The information gathered from a device is first transformed using a function, so it can be processed. After this is done it is necessary to undo the transformation using an inverse function. This processing can be finding a derivative, and sometimes it is even possible to work on the derivative of the inverse function itself! In this article, we will take a look at how this is done.

Rule for the Derivative of Inverse Functions

If you know the derivative of a function you can find the derivative of its inverse without using the definition of a derivative. Here is how you can do it.

Let \( f(x) \) be an invertible and differentiable function, and let \( f^{-1}(x) \) be its inverse. If \( f^{-1}(x) \) is differentiable, its derivative is given by the following formula:

$$\left( f^{-1} \right)' (x) = \frac{1}{f'\left( f^{-1}(x) \right)}.$$

This means that you need to find the derivative of \( f(x) \) and find its composition with \( f^{-1}(x). \) Assuming that \( f^{-1}(x) \) is known, this procedure can be summarized in the following steps:

1. Find the derivative of \( f(x) \), that is, find \( f'(x). \)

2. Find the composition \( f' \left( f^{-1}(x) \right). \)

3. Take the reciprocal of \( f' \left( f^{-1}(x) \right). \)

This is better understood by looking at some examples.

Derivatives of Inverse Functions Examples

There is a wide variety of invertible functions that we can differentiate, so let's take a look at some examples.

Derivatives of Irrational Functions

Square root functions and quadratic functions are inverses of each other. You can find the derivative of a quadratic function by using the Power Rule, and then, use this result to find the derivative of a square root function.

Derivatives of Logarithmic Functions

Even though you can find the derivative of logarithmic functions using the definition of a derivative, you can also use the fact that the logarithmic function is the inverse of the exponential function.

This procedure is an excellent alternative to finding the derivative of the natural logarithmic function using the definition of a derivative!

Common Mistakes When Finding the Derivative of an Inverse Function

There are two common mistakes when finding the derivative of an inverse function.

• Doing the composition in the wrong order.

• Forgetting to take the reciprocal of the composition.

Let's take a look at each.

Doing the composition in the wrong order

One common mistake is getting the composition in the wrong order. Remember that, in general

$$f'\left( f^{-1}(x)\right) \neq f^{-1}\left( f'(x) \right).$$

Let's see an example of a composition done in the wrong order.

Forgetting to take the reciprocal of the composition

Another common mistake is forgetting to take the reciprocal after finding the composition. That is

$$\left( f^{-1} \right) ' (x) \neq f'\left( f^{-1}(x)\right),$$

so always remember to take the reciprocal of the composition

$$\left( f^{-1} \right) ' (x) = \frac{1}{f'\left( f^{-1}(x)\right)}.$$

Derivative of an inverse function from a table

Functions can be represented in many ways. Take for instance a table of values.

You can use the above table to find the slope of a line secant to the function by taking two points an applying the slope formula

\[ m=\frac{y_2-y_1}{x_2-x_1}\]

as shown in the following image.

Fig. 1. Graph of a line secant to the function at two points.

In the above image the points \( (1,f(1)) \) and \( (2,f(2)) \) are being used for the secant line, so its slope is

\[ \begin{align} m &= \frac{f(2)-f(1)}{2-1} \\ &= \frac{4-1}{1} \\ &= 3. \end{align} \]

By switching the values of the table you can get the inverse of the original function, in this case you will get the square root function.

You can also find a line secant to the square root function by using two points. The first point can stay the same as it is shared amongst both functions. For the second point rather than using \( (2,f^{-1}(2)) \) you have compose the function, that is, you need to use \( (f(2),f^{-1}(f(2))).\) Therefore the corresponding secant uses \( (1,f^{-1}(1)) \) and \( (4,f^{-1}(4)) \) instead as shown in the following image.

Fig. 2. Graph of a line secant to the inverse function at two points.

This time the points \( (1,f^{-1}(1)) \) and \( (4,f^{-1}(4)) \) are being used for the secant line, so its slope is

\[ \begin{align} m &= \frac{f^{-1}(4)-f^{-1}(1)}{4-1} \\ &= \frac{2-1}{3} \\ &= \frac{1}{3}. \end{align} \]

Note that the slope of the secant of the inverse function is the reciprocal of the slope of the line secant to the original function. Furthermore, you need to compose the function and the inverse to find the above slope. Do these steps sound familiar?

The slope of secant lines are related to derivatives by means of a limit. The above reasoning keeps working as you take smaller intervals, connecting to the formula of the derivative of an inverse function.

Proof of the Derivative of an Inverse Function Formula

The proof of the Derivative of an Inverse Function uses the fact that the composition of a function and its inverse is equal to the identity function, that is

$$f\left(f^{-1}(x)\right)=x.$$

Next, differentiate both sides of the equation. You use the Power Rule to differentiate the right-hand side of the equation, so

$$\frac{\mathrm{d}}{\mathrm{d}x} f\left(f^{-1}(x)\right) = 1.$$

The left-hand side of the equation can be differentiated using the Chain Rule, giving you

$$f'\left(f^{-1}(x)\right) \left( f^{-1} \right) ' (x)=1,$$

and finally, you can isolate the derivative of the inverse function

$$\left( f^{-1} \right)'(x)=\frac{1}{f'\left( f^{-1}(x) \right)}.$$