What is inverse of $f(x)$ when $f(x)= \int_{2}^{x} \ln(t) \,dt $

Question

Given that $f(x)= \int_{2}^{x} \ln(t) \,dt $. I have to find $(f^{-1})'(0)$. If I know $f^{-1}$ I may find $(f^{-1})'(0)$. So I am trying to find $f^{-1}$ but I am stuck.

Answer 1

Let me briefly reiterate the derivative-part of the Inverse Function Theorem: The derivative of the inverse is given by

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

First of all, observe that $f(2) = \int_2^2 \ln(t) \, dt = 0.$ Hence,

$$(f^{-1})'(0) = \frac{1}{f'(2)}.$$

We find $f'(2)$ using the Fundamental Theorem of Calculus and we get that $f'(2) = \ln(2)$. Thus,

$$(f^{-1})'(0) = \frac{1}{\ln(2)}.$$