91Ó°ÊÓ

Finding the derivative of an inverse function is a crucial skill in calculus. It helps us understand how the rate of change of one variable with respect to another is mirrored when swapped. Let's see how this works by revisiting the inverse function of the original function given, which is
\[ f(x) = x^3 + 3 \].
The inverse is represented by \( f^{-1}(x) = (x - 3)^{\frac{1}{3}} \). To find the derivative of \( f^{-1}(x) \), we often use the formula:

• If \( y = f^{-1}(x) \), then the derivative is \( \frac{1}{f'(f^{-1}(x))} \).

In our case, the derivative \( \frac{d}{dx} \) for \( f(x) = x^3 + 3 \) originally is \( 3x^2 \).
We find \( f'(f^{-1}(x)) \) by substituting \( f^{-1}(x) \), giving us:
\[ f'(f^{-1}(x)) = 3((x - 3)^{\frac{1}{3}})^2 \].
Thus, the derivative of the inverse function becomes:
\[ \frac{1}{3((x - 3)^{\frac{1}{3}})^2} = \frac{1}{3} (x - 3)^{-\frac{2}{3}} \].
This expression provides the rate at which the inverse function's values change as \( x \) changes.

The chain rule is an essential tool in calculus for differentiating compositions of functions. It allows us to break down complex expressions into manageable parts. In the context of inverse functions,
it becomes especially useful when differentiating nested relationships.
The chain rule states:

• If a function \( y = f(g(x)) \), its derivative \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

In the problem context, we've rewritten the inverse function as a composition, \( f^{-1}(x) = u^{\frac{1}{3}} \) where \( u = x - 3 \).
When we differentiate,

• First, differentiate \( u^{\frac{1}{3}} \) with respect to \( u \): \( \frac{1}{3} u^{-\frac{2}{3}} \).
• Then, differentiate \( u \) = \( x - 3 \) with respect to \( x \): 1.

Using the chain rule, the derivative of the inverse function becomes:
\[ \frac{d}{du}(u^{\frac{1}{3}}) \cdot \frac{d}{dx}(x - 3) = \frac{1}{3} (x - 3)^{-\frac{2}{3}} \].
This demonstrates how the chain rule connects the changes in and out of the function.

To find an inverse function, we often solve an equation for the dependency variable. For instance, given a function \( y = f(x) \), finding its inverse \( f^{-1}(x) \) involves reversing the roles of \( x \) and \( y \).
Let's break this down with the function \( f(x) = x^3 + 3 \):

• Initially, we let \( y = x^3 + 3 \) to represent \( f(x) \).
• Next, switch \( x \) and \( y \) to form \( x = y^3 + 3 \) to initiate the inversion process.
• Finally, solve for \( y \) to express \( y = (x - 3)^{\frac{1}{3}} \).

This process unveils the inverse function \( f^{-1}(x) \), allowing us to express \( x \) in terms of \( y \), making the original inputs become outputs.
Understanding this approach is key in navigation through calculus and discovering how to switch the dependencies of the equations effortlessly.