This chapter is devoted to other topics involving differential calculus that don’t fit into a specific
category.
THE DERIVATIVE OF AN INVERSE FUNCTION
ETS occasionally asks a question about finding the derivative of an inverse function. To do this, you need
to learn only this simple formula.
Suppose we have a function x = f(y) that is defined and differentiable at y = a where x = c. Suppose we
also know that the f −1(x) exists at x = c. Thus, f(a) = c and f −1(c) = a. Then, because ,
To translate, we can find the derivative of a function’s inverse at a particular point by taking the
reciprocal of the derivative at that point’s corresponding y-value. These examples should help clear up
any confusion.
Example 1: If f(x) = x^2 , find a derivative of f −1(x) at x = 9.
First, notice that f(3) = 9. One of the most confusing parts of finding the derivative of an inverse function
is that when you’re asked to find the derivative at a value of x, they’re really asking you for the derivative
of the inverse of the function at the value that corresponds to f(x) = 9. This is because x-values of the
inverse correspond to f(x)-values of the original function.
The rule is very simple: When you’re asked to find the derivative of f −1(x) at x = c, you take the
reciprocal of the derivative of f(x) at x = a, where f(a) = c.
We know that f(x) = 2x. This means that we’re going to plug x = 3 into the formula (because f(3) = 9).
This gives us