Derivatives of Inverse Functions

Inverse Functions:

Inverse functions can be written as f^(-1)(x) and they occur when the y and x values of a function are switched so that the graph flips over dotted line in the example below.



Finding the Derivative of an Inverse Function:

The derivative of an inverse function is the same as the reciprocal of the derivative of the original function when the x value of the inverse function is equal to the y value of the function. In other words, to find the derivative of the function you must:

1. Find the x value of the original function so that the y value original function is equal to the x value of the inverse function.
2. Find the derivative of the original function when x is equal to the x value you found in step 1.
3. Find the derivative of the inverse function by finding the reciprocal of the answer from step 2. (In other words flip the fraction, but remember that the sign does not change)

Example:
Find (f^-1)'(3) when f(x) = -x^2+5

3 = x+5
5-3 = x
2 = x
f'(x) = -2x^1+0 = -2x
f'(2) = -2(2) = -4
(f^-1)'(3) = -1/4

Sometime you are given a chart instead of an equation.

Example:

Find (g^-1)'(3)

3 = g(x)
x = 4
g'(4) = 1/2
(g^-1)'(3) = 2