Question

Suppose f is a one-to-one function with f(2)=8 and f’(2)=4. What is the value of (f^-1)’(8)?

Answer 1

1/4

Explanation:

The tangent line to f(x) at x=2 goes through the point (2, 8) and has slope 4.

The inverse function f⁻¹(x) is a reflection of the function f(x) across the line y=x. The corresponding tangent line will go through point (8, 2) and have slope 1/f'(2) = 1/4 at that point. That is, ...

(f⁻¹)'(8) = 1/f'(f⁻¹(8)) = 1/f'(2) = 1/4