Let's say $f:D\to R$ is an injective function on some domain where it is also differentiable. For a real function, i.e. $D\subset\mathbb R, R\subset\mathbb R$, is it possible that $f'(x)\equiv f^{-1}(x)$?
Intuitively speaking, I suspect that this is not possible, but I can't provide a reasonable proof since I know very little nothing about functional analysis. Can anyone provide a (counter)example or prove that such function does not exist?
Best Answer
It is possible! Here is an example on the domain $D=[0,\infty)$: $$ f(x) = \bigg(\frac{\sqrt{5}-1}{2}\bigg)^{(\sqrt5-1)/2} x^{(\sqrt5+1)/2}. $$ I found this by supposing that $f(x)$ had the form $ax^b$, setting the derivative equal to the inverse function, and solving for $a$ and $b$.