Introduction to the Derivative of Inverse Functions

In calculus, there are different and easy ways to calculate derivatives of a function by using simple differentiation rules. But sometimes we need to calculate derivatives of an inverse function for which simple derivative rules may not work. Let’s understand how to calculate derivative of an inverse function and the difference between inverse function and implicit differentiation. 

Understanding of the Derivative of inverse functions

The inverse function differentiation is a method to calculate derivative of an inverse function. This method is suitable to calculate the rate of change of a function which is invertible and differentiable. An invertible function is a function whose inverse exists. If an invertible function is differentiable then the inverse of this function is also differentiable. 

In inverse function differentiation, we calculate the derivative of inverse. But since there is no specific formula to calculate derivative of such functions. Therefore, the derivative of the original function is used. The relation between a function and its inverse helps to calculate the inverse derivative. By using this differentiation, we can avoid the use of limit definition of derivative. 

Derivative of Inverse Function Formula

The inverse function derivative formula can be obtained by using the relation between a function with its inverse. For this assume an inverse function which is expressed as:

$y=f^{-1}(x)$

Since the function is invertible, so we can write it as,

$x = f(y)$

Now the function becomes an implicit function. We can use implicit differentiation to solve this. Applying derivative, 

$\frac{d}{dx}(x)=\frac{d}{dx}[f(y)]$

$1=f'(y)\frac{dy}{dx}$

Or, 

$\frac{dy}{dx}=\frac{1}{f'(y)}$

But here $y = f^{-1}(x)$, therefore, 

$\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'[f^{-1}(x)]}$

Or, 

$[f^{-1}]'(x)=\frac{1}{f'[f^{-1}(x)]}$

This formula is suitable for inverse trigonometric differentiation which calculates the derivative of inverse trig functions. 

How to calculate derivatives of inverse functions?

The implementation of implicit differentiation is divided into a few steps. These steps assist you to calculate the derivative of an implicit function. These steps are:

1. Write the expression of the function. 
2. Identify the independent and dependent variables. 
3. Write the inverse function as an explicit function so that it can be differentiated easily. 
4. You can also simply use the inverse function differentiation formula. 
5. Apply the derivative on both sides of the equation with respect to the independent variable. 
6.  For example if the left side of the equation contains a power function, use the power rule derivative formula.
7. In the final step, simplify the equation and rearrange it to get dy/dx.

Let’s understand the following examples to find derivative of inverse functions.

Derivative of inverse function example 1

Find the derivative of inverse trigonometric function tan inverse by using inverse derivative formula.

To calculate the derivative of arctan, suppose that,

$y = \tan^{-1}x$

It can be written as,

$\tan y = x$

Now applying derivative on both sides,

$\frac{d}{dx}(\tan y) = \frac{d}{dx}(x)$

Since the derivative of tan is equal to sec square and the derivative of x is 1.

$\sec^2 y \frac{dy}{dx} = 1$

Rearranging the equation, 

$\frac{dy}{dx} = \frac{1}{\sec^2y}$

Since $y = \tan^{-1}x$ then, 

$\frac{d}{dx}(\tan^{-1}x) =\frac{1}{\sec^2(\tan^{-1}x)}$

Assume that tan-1x = θ then tan θ = x. Constructing a triangle with angle θ where adjacent side is 1, opposite side is x and then calculating hypotenuse by using Pythagorean Theorem which is √1+x2. Therefore, 

$\sec^2(\tan^{-1}x)=\sec^2\theta =(\sqrt{1+x^2})^2 = 1+x^2$

Using the above value in the derivative of tan-1x.

$\frac{d}{dx}(\tan^{-1}x) = \frac{1}{1+x^2}$

Derivative of inverse function example 2

Find the derivative of inverse trigonometric function sine inverse by using inverse derivative formula.

To calculate the derivative of arcsin, suppose that,

$y=\sin^{-1}x$

It can be written as,

$\sin y = x$

Now applying derivative on both sides,

$\frac{d}{dx}[\sin y] = \frac{d}{dx}(x)$

Since the derivative of sin is equal to cos square and the derivative of x is 1.

$\cos y\frac{dy}{dx}=1$

Rearranging the equation, 

$\frac{dy}{dx}=\frac{1}{\cos y}$

Since $y = sin^{-1}x$ then, 

$\frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sin(\sin^{-1}x)}$

Assume that $\sin^{-1}x = \theta$ then $\sin\theta = x$. Constructing a triangle with angle θ where adjacent side is 1, opposite side is x and then calculating hypotenuse by using Pythagorean Theorem which is √1-x2. Therefore, 

$\cos(\sin^{-1}x)=\cos\theta = \sqrt{1-x^2}$

Using the above value in the derivative of sin-1x.

$\frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}}$

Calculating derivative of inverse functions by using calculator

The derivative of an inverse function can be also calculated by using an inverse function derivative calculator. It is an online tool that follows the inverse differentiation formula to find derivative. You can find it online by searching for a derivative calculator. For example, to calculate the derivative of cos inverse, the following steps are used by using this calculator.

1. Write the expression of the function in the input box such as, cos^-1 x.
2. Choose the variable to calculate the rate of change, which will be x in this example.
3. Review the input so that there will be no syntax error in the function. 
4. Now at the last step, click on the calculate button. By using this step, the inverse function derivative calculator will provide the derivative of cos inverse quickly and accurately which will be -1/√1-x2.

Comparison between derivative of inverse function and implicit differentiation

The comparison between the derivative of inverse function and implicit differentiation can be easily analysed using the following difference table.

Conclusion

The derivative of an inverse function is a fundamental way to calculate the derivative of a function. But to find an inverse derivative, a function should be both differentiable and invertible. The relation between a function and its inverse is used to find an inverse function derivative.