# Question Video: Finding the Differentiation of a Given Inverse Function Mathematics • Higher Education

If 𝑓(2𝜋) = −1, 𝑓′(2𝜋) = 1, and 𝑎 = −1, find (𝑓⁻¹)′ (𝑎).

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### Video Transcript

If 𝑓 of two 𝜋 is equal to negative one, 𝑓 prime of two 𝜋 is equal to one, and 𝑎 is equal to negative one, find the derivative of the inverse of 𝑓 at 𝑎.

We’ll be using the fact that the derivative of the inverse of 𝑓 at 𝑎 is equal to one over 𝑓 prime of 𝑓 inverse of 𝑎. In our case, 𝑎 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 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. 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. And we can see that we’ve actually been given 𝑓 prime of two 𝜋 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.