Given the function
$$ f(x) = e^x + \arctan(x) = y\;, $$
what is the inverse $f^{-1}(y)=\dots\;$, and how can I find it? I’m looking for solutions including all steps and possible explanations along with each.
To give some wider context, I bumped into this problem as part of a bigger question asking me to prove that $f(x)$ is bijective, $(f^{-1})'(y)$ exists for all $y>-\pi/2$ and to calculate $(f^{-1})'(y)$.
I have proven that $f$ is bijective and the rest of the properties follow from the applicability of the inversion theorem for derivatives, but I have a hard time calculating $(f^{-1})'(y)$ now because I can't find $(f^{-1})(y)$
Thanks for your help!