Function of a Function Differential Equation

Question

Is there any function, $f(x)\neq x$, for which $f(f'(x))=f'(f(x))$?

Answer 1

For the real-valued function $f(x)=e^x$, $f'(x)=e^x$.

Answer 2

Why, many. $f(x)={x^2\over2}$ will do.

If you want to exclude the "trick" answers where $f'(x)=x$ or $f(x)=f'(x)$, go with $f(x)={x^3\over9}$. Other examples are still plenty, I believe.

Upd. The other examples seem less numerous than I initially believed, but anyway, $f(x)=\dfrac{x^n}{n^{n-1}}$ for any constant $n$ is good.