$$ f^{-1}(x)=f'(x)$$ Where $f^{-1}:$ Inverse of f
I know for a fact there is a specific solution that is demonstrated on this video and specifically:
$$ f(x) = x^{φ}\sqrt[φ]{\frac{1}{φ}} $$ Where $φ=\frac{1+\sqrt{5}}{2}$ is the golden ration.
What I wish to do, is to find (if they exist) the rest of the solutions to the equation. In case there is no other, I want to at least prove that there is no other solution. For the record, I seek a conventional solution with a mathematical proof backing it, and not just through making "educated" guesses.