Video: Applying the Inverse Function Theorem

If 𝑓(5) = 3, 𝑓′(5) = 1/4, and π‘Ž = 3, find (𝑓⁻¹)β€²(π‘Ž).

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Video Transcript

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

To answer this question, we’re going to apply the inverse function theorem. This says let 𝑓 of π‘₯ be a function that’s both invertible and differentiable. We say that 𝑦 equals 𝑓 superscript negative one of π‘₯ is the inverse of 𝑓 of π‘₯. For all π‘₯ satisfying 𝑓 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 𝑓 prime of the inverse of π‘₯.

So how are we going to use this to evaluate the derivative of the inverse of 𝑓 evaluated at π‘Ž? Well, we’re told that π‘Ž 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 π‘Ž is four.

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