How to Find the Derivative of the Inverse Function: An Illustrative Example with Step-by-Step Guide

Derivative of Inverse

To find the derivative of the inverse of a function, we can use the inverse function theorem

To find the derivative of the inverse of a function, we can use the inverse function theorem. The inverse function theorem states that if a function f(x) has an inverse function f^(-1)(x), and if f'(x) is continuous and nonzero in a neighborhood of a point a, then the derivative of the inverse function f^(-1)(x) at the point b = f(a) is given by:

(f^(-1))'(b) = 1 / f'(a)

This means that the derivative of the inverse function at a point b is equal to the reciprocal of the derivative of the original function at the corresponding point a.

Let’s go through an example to illustrate this concept. Consider the function f(x) = 3x + 2. We want to find the derivative of its inverse function.

Step 1: Find the inverse of f(x):
To find the inverse function, we need to swap the roles of x and y and solve for y. Let’s call the inverse function g(x):

x = 3y + 2
x – 2 = 3y
y = (x – 2) / 3

Therefore, the inverse function of f(x) = 3x + 2 is g(x) = (x – 2) / 3.

Step 2: Find the derivative of f(x):
To find the derivative of f(x), we simply differentiate it with respect to x:

f'(x) = 3

Step 3: Find the derivative of g(x):
Using the inverse function theorem, we know that:

(g^(-1))'(b) = 1 / f'(a)

Since we found that the inverse function g(x) is (x – 2) / 3, we can see that:

(g^(-1))'(b) = 1 / f'(a) = 1 / 3

Therefore, the derivative of the inverse function g(x) = (x – 2) / 3 is:

(g^(-1))'(x) = 1 / 3

To summarize, the derivative of the inverse function is constant and equal to the reciprocal of the derivative of the original function.

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