Given $f(x)$ with inverse $g$, find $g'(2)$

Question

Problem

Given the function $$f(x) = \ln^3(x) - 2\ln^2(x) + \ln(x)$$ defined for $$x\in[e, e^3]$$ show that the function has an inverse $g$ on the given interval, and find $g'(2)$

Progress

I have concluded that $f$ has an inverse on the interval by showing that it is continuous and strictly increasing.

Question

How do I find $g'(2)$? I have contemplated trying to calculate the inverse of $f$ by hand, but given the wording of the problem, I don't think that's intended.

Can I find $g'(2)$ using some other neat trick?

Any help/solution appreciated!

Answer 1

$g'(2)={1\over f'(x_0)}$, where $x_0$ is such that $f(x_0)=2$. Can you find it?

Answer 2

Hint:

$$g'(x) = \frac{1}{f'(g(x))}$$

That could help.