Differentiation of Inverse Functions

Neetesh Kumar | February 08, 2025 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Definition of Derivative of Inverse Functions
• 2. Formula and Concept
• 3. Solved Examples
• 4. Conclusion

Derivative of Inverse Functions:
Let $f(x)$ be a function with an inverse $f^{-1}(x)$. The derivative of the inverse function $f^{-1}(x)$ is given by:

$\boxed{\dfrac{d}{dx} \bigg(f^{-1}(x)\bigg) = \frac{1}{f'\bigg(f^{-1}(x)\bigg)}}$

This formula tells us that the derivative of an inverse function is the reciprocal of the derivative of the original function evaluated at the point $f^{-1}(x)$.

Example 1: Differentiating the Inverse of a Polynomial Function

Find the derivative of the inverse function of $f(x) = 2x^3 - 3x$.

Solution:

First, differentiate $f(x)$:

$f'(x) = 6x^2 - 3$.

The inverse function is $f^{-1}(x)$. According to the formula:

$\dfrac{d}{dx} \bigg(f^{-1}(x)\bigg) = \dfrac{1}{f'(f^{-1}(x))}$.

Now, let $y = f^{-1}(x)$, then:

$x = 2y^3 - 3y$

We need $f'(y)$, so substitute $y = f^{-1}(x)$ into the derivative of $f(x)$:

$f'\bigg(f^{-1}(x)\bigg) = 6(f^{-1}(x))^2 - 3$.

Thus, the derivative of the inverse function is:

$\boxed{\dfrac{d}{dx} \bigg(f^{-1}(x)\bigg) = \frac{1}{6(f^{-1}(x))^2 - 3}}$.

Example 2: Inverse of the Exponential Function

Find the derivative of the inverse function of $f(x) = e^{2x}$.

Solution:

First, differentiate $f(x)$:

$f'(x) = 2e^{2x}$.

The inverse function is $f^{-1}(x) = \dfrac{1}{2} \ln x$, so applying the derivative formula:

$\dfrac{d}{dx} \bigg(f^{-1}(x)\bigg) = \dfrac{1}{f'(f^{-1}(x))}$

Substitute $f^{-1}(x) = \frac{1}{2} \ln x$ into $f'(x)$:

$f'(f^{-1}(x)) = 2e^{2 \cdot \frac{1}{2} \ln x} = 2x$.

Thus, the derivative of the inverse function is: $\boxed{\dfrac{d}{dx} \bigg(f^{-1}(x)\bigg) = \frac{1}{2x}}$.

Example 3: Differentiating the Inverse of a Logarithmic Function

Find the derivative of the inverse function of $f(x) = \ln(x^2 + 1)$.

Solution:

First, differentiate $f(x)$:

$f'(x) = \dfrac{2x}{x^2 + 1}$.

The inverse function is $f^{-1}(x)$, and applying the chain rule for inverse functions:

$\dfrac{d}{dx} \bigg(f^{-1}(x)\bigg) = \dfrac{1}{f'(f^{-1}(x))}$

Substitute into the derivative of $f(x)$:

$f'(f^{-1}(x)) = \dfrac{2f^{-1}(x)}{(f^{-1}(x))^2 + 1}$.

Thus, the derivative of the inverse function is:

$\boxed{\dfrac{d}{dx} \bigg(f^{-1}(x)\bigg) = \frac{(f^{-1}(x))^2 + 1}{2f^{-1}(x)}}$.

Example 4: Inverse of the Sine Function

Find the derivative of the inverse sine function $f(x) = \sin^{-1}(x)$.

Solution:

The derivative of the inverse sine function is known to be:

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

Thus, the derivative of the inverse sine function is:

$\boxed{\dfrac{d}{dx} \bigg(\sin^{-1}(x)\bigg) = \dfrac{1}{\sqrt{1 - x^2}}}$.