Differential of strict function

Question

Let $e^{f(x) } $=lnx.if g)(x) is the inverse of f(x) , the n derivative of g(x) is

I first found the inverse of the function which is $e^{e^x}$ and then substituted it on the formulae
$$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$$ but the answer I am getting is not matching. I got x$e^{e^x}$ which is not correct. How to do this

Answer 1

$$e^{f(x)}=\ln x$$ $$f(x)=\ln(\ln x)$$ $$y=\ln(\ln x)$$

swap $x$ and $y$ $$x=\ln(\ln y)$$

thus $$f^{-1}(x)=\ln(\ln x)$$

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