User:TigerTjäder/Math

Suppose $$f^{-1}$$ is the inverse function of a differentiable function $$f$$ and $$f(4) = 5$$, $$f'(4) = 2/3$$. Find $$(f^{-1})'(5)$$

We are gonna use two identities:

(1) $$(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}$$

and

(2) $$f(x) = y \iff f^{-1}(y) = x$$

By (2) and $$f(4) = 5$$ we have (3) $$f^{-1}(5) = 4$$.

Using (1) and setting $$a = 5$$, we have

(4) $$(f^{-1})'(5) = \frac{1}{f'(f^{-1}(5))}$$

Applying (3) on (4), we get

(5) $$(f^{-1})'(5) = \frac{1}{f'(4)}$$

The problem says $$f'(4) = 2/3$$, so we substitute it on (5):

$$(f^{-1})'(5) = \frac{1}{\frac{2}{3}}$$

After some manipulation:

$$(f^{-1})'(5) = \frac{3}{2}$$