Absolutely there is a trivial solution $ f(x)=0$ . Actually, assuming $ f(x)$ being smooth and expanding $ f(x)$ into power series one can get $ f(0)=0\to f(x)=0$ . Also, in the complex field there are solutions e.g. $ f(x)=(-\omega)^{\left(-\omega^2\right)}x^{-ω}$ where $ \omega=e^{\pm 2\pi i/3}$ . So I am wondering, are there any non-zero $ f:\mathbb{R}\to \mathbb{R}$ such that $ f(f(x))=f'(x)$ ?

If there is no solution on $ \mathbb{R}$ , then are there any solutions on some real interval, such as $ f:[a,b]\to[a,b]$ ?