Video: Differentiation for Inverse Functions

If 𝑓(1/2) = βˆ’6, 𝑓′(1/2) = 3, and π‘Ž = βˆ’6, find (𝑓⁻¹)β€²(π‘Ž).

01:31

Video Transcript

If 𝑓 of a half equals negative six, 𝑓 prime of a half equals three, and π‘Ž equals negative six, find the inverse of 𝑓 prime of π‘Ž.

Let’s start by recalling a useful formula for finding the derivative of an inverse function. That is the inverse of 𝑓 prime of π‘Ž equals one over 𝑓 prime of the inverse of 𝑓 of π‘Ž. For our question, π‘Ž equals negative six. So we can substitute this into the formula. So we’re going to need to find what 𝑓 prime of the inverse of 𝑓 of negative six is. Well, we can find what the inverse of 𝑓 of negative six is using the information in the question and the definition of the inverse function.

The definition of the inverse function says that 𝑓 of the inverse of 𝑓 of π‘Ž equals π‘Ž. And so 𝑓 of the inverse of 𝑓 of negative six equals negative six. From the information given in the question, 𝑓 of one-half equals negative six. So the inverse of 𝑓 of negative six must be equal to a half. If we substitute this back into our workings, we find that the inverse of 𝑓 prime of negative six equals one over 𝑓 prime of one-half. From the question, we’re told that 𝑓 prime of one-half equals three. And so we find that the inverse of 𝑓 prime of negative six equals one over three.

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