3.8 Derivatives of Inverse Functions

At x = 2: We can find the inverse function as follows: To find the derivative of the inverse function: Switch x and y.

Slopes are reciprocals. At x = 2: At x = 4:

Slopes are reciprocals. Because x and y are reversed to find the reciprocal function, the following pattern always holds: The derivative of Derivative Formula for Inverses: evaluated at is equal to the reciprocal of the derivative of evaluated at .

A typical problem using this formula might look like this: Given: Find: Derivative Formula for Inverses:

Let’s try slightly different notation: In other words, g is the inverse of f If at the point (a,b) Then at the point (b,a) Slopes are reciprocals. Onward…

We know that and that But what about… Let’s try writing the function like this: and taking the natural log of both sides…

This is called Logarithmic Differentiation p