TITLE: Examples of functions where $f'(x)=f(f(x))$ for all $x$
QUESTION [7 upvotes]: I am looking for examples of functions $f:\mathbb R \to \mathbb R$ where $f'(x)=f(f(x))$ for all $x$. The only example I can find is the trivial one where f is identically 0.

REPLY [5 votes]: This has been solved here:

Result: for every $a > 0$, there exists a unique differentiable function $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(-a)=-a$ and $f' = f \circ f$.