If $f(0)=2$, $f'(0)=3$ and $g=f^{-1}$ then what is the value of $g'(2)$?

Question

I know that $f(0)=2$, $f'(0)=3$ and $g=f^{-1}$.

But how can I find the value of $g'(2)$?

Answer 1

You have: $g(f(x))=x$

Differentiate both sides: $g' (f(x)) f'(x)=1$
Put $x=0$ $g' (f(0)) f'(0) =1$
Therefore,$ g'(2)= 1/3$

Answer 2

Use the fact that $g \circ f = Id$, and derive both sides in 0, you get: $$ f'(0) \cdot g'(f(0)) = 1 \\ \textrm{thus }g'(f(0)) = \frac{1}{f'(0)} $$ Hence, $g'(2) = \frac{1}{3}$ .

Btw that's how you get the derivative of a reciprocal function in general.