Is there a formula for the inverse of a derivative?

Question

Given a function $f$ with the derivative $f'$ is there a general formula to deduce $(f')^{-1}$ provided it exists?

Answer 1

No. Take $f(x)=e^{e^x}$. We have that $f^{-1}=\ln(\ln(x))$. If there were a formula for $(f')^{-1}$, then it would be an elementary function. However, $f'(x)=e^{e^x+x}$, whose inverse is not an elementary function. In fact, its inverse is $(f')^{-1}(x)=\ln(x)-W(x)$, where $W$ is the Lambert W function, defined to be the inverse of $xe^x$. This function is known not to be elementary, and thus $(f')^{-1}=\ln(x)-W(x)$ is not elementary.