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.
