Video: Applying the Inverse Function Theorem

Video Transcript

If 𝑓 of five is equal to three, 𝑓 prime of five is equal to one-quarter, and 𝑎 equals three, find the derivative of the inverse of 𝑓 evaluated at 𝑎.

To answer this question, we’re going to apply the inverse function theorem. This says let 𝑓 of 𝑥 be a function that’s both invertible and differentiable. We say that 𝑦 equals 𝑓 superscript negative one of 𝑥 is the inverse of 𝑓 of 𝑥. For all 𝑥 satisfying 𝑓 prime of the inverse of 𝑥 is not equal to zero, we can say that the derivative of the inverse of 𝑥 is equal to one over 𝑓 prime of the inverse of 𝑥.

So how are we going to use this to evaluate the derivative of the inverse of 𝑓 evaluated at 𝑎? Well, we’re told that 𝑎 equals three. So we know we want the derivative of the inverse evaluated at three. And according to the inverse function theorem, that must be equal to one over 𝑓 prime of the inverse at three.

We’re also told that 𝑓 of five equals three. Now, this, in turn, must mean that the inverse of 𝑓 at three must be equal to five. And so we replace this in the inverse function theorem. And we see that the derivative of the inverse of 𝑓 evaluated at three is one over 𝑓 prime of five. But we’re also told that 𝑓 prime of five is equal to one-quarter. So this becomes one over one-quarter, which is simply four.

And so the derivative of the inverse of 𝑓 evaluated at 𝑎 is four.