### Video Transcript

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

To answer this question, weβre going to apply the inverse function theorem. This says let π of π₯ be a function thatβs both invertible and differentiable. We say that π¦ equals π superscript negative one of π₯ is the inverse of π of π₯. For all π₯ satisfying π prime of the inverse of π₯ is not equal to zero, we can say that the derivative of the inverse of π₯ is equal to one over π prime of the inverse of π₯.

So how are we going to use this to evaluate the derivative of the inverse of π evaluated at π? Well, weβre told that π 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 π prime of the inverse at three.

Weβre also told that π of five equals three. Now, this, in turn, must mean that the inverse of π 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 π evaluated at three is one over π prime of five. But weβre also told that π 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 π evaluated at π is four.