Video: Finding the Differentiation of a Given Inverse Function

If 𝑓(2πœ‹) = βˆ’1, 𝑓′(2πœ‹) = 1, and π‘Ž = βˆ’1, find (𝑓⁻¹)β€² (π‘Ž).

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Video Transcript

If 𝑓 of two πœ‹ is equal to negative one, 𝑓 prime of two πœ‹ is equal to one, and π‘Ž is equal to negative one, find the derivative of the inverse of 𝑓 at π‘Ž.

We’ll be using the fact that the derivative of the inverse of 𝑓 at π‘Ž is equal to one over 𝑓 prime of 𝑓 inverse of π‘Ž. In our case, π‘Ž is equal to negative one. We need to start by finding 𝑓 inverse of negative one. We’re given in the question that 𝑓 of two πœ‹ is equal to negative one. Since we know that 𝑓 inverse is the inverse function of 𝑓, this tells us that 𝑓 inverse of negative one is equal to two πœ‹. So we can substitute this into our equation. And now, we have that the derivative of 𝑓 inverse at negative one is equal to one over 𝑓 prime of two πœ‹. And we can see that we’ve actually been given 𝑓 prime of two πœ‹ in the question. And it’s equal to one.

So we can substitute this in. And we’ve reached our solution, which is that the derivative of the inverse function of 𝑓 at negative one is equal to one.

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