Example of a function that its derivative is the same as its inverse. That is
$$f'(x)=f^{-1}(x)$$
In this video Michael Penn, shows a technique and presents a function with that property. The example is
$$f(x)=\sqrt[\varphi]{\dfrac{1}{\varphi}} \cdot x^{\varphi}$$ where $\varphi=\dfrac{1+\sqrt{5}}{2}$
But his technique is based on an intelligent guess, is it possible to solve this problem in another way and find another function with this property?