When could we get $f' = f^{-1}$, where $f^{-1}$ is the inverse function of $f$, and not $\frac{1}{f}$

Personal question : Is there a general solution to the equation $f' = f^{-1}$, where $f^{-1}$ is the inverse function of $f$, and not $\frac{1}{f}$.

I think this question is difficult, and I don't have the competence to answer it.

Are there someone who is able to rigorously answer this question?

I don't think any such function is possible, at least if the domain of $f$ is supposed to be all of $\mathbb{R}$. If $f^{-1}$ is defined, then $f$ is injective, and if $f$ is also continuous, this means that $f$ is monotone, so either $f'(x) \geq 0$ for all $x$ or $f'(x) \leq 0$ for all $x$. Either way, $f'$ is not surjective, but $f^{-1}$ must be surjective (since $f$ is defined on all of $\mathbb{R}$), so $f' \neq f^{-1}$.