Find the derivative of $f^{-1}(x)$ at $x=2$ if $f(x)=x^2 + x + \ln x$

Question

I'm fairly confused with this question (or I guess the concept of inverse functions and taking their derivative). I know that the general rule for taking the derivative of an inverse function is:

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

But I'm not really sure where to go from here. A nudge in the right direction would be greatly appreciated.

Answer 1

First : $f(1)=1+1+0=2$ and if you derivate f you can easily show that it is strictly positive (i.e. increasing) so $1$ is the only value such that $f(1)=2$.

so

$$f^{-1'}(2) = \frac{1}{f'(f^{-1}(2))}= \frac{1}{f'(1)}=\frac{1}{2\cdot1+1+\frac{1}{1}}=\frac{1}{4}$$