How can we visualize the orbits of the derivative?

Question

The derivative is a function of functions with fixed points at $e^x$, $ce^x$ for constant $c$, and $0$. Is there any possible way we can visualize its orbits, even for just a subset of all functions (e.g. polynomials)?

Answer 1

There's not just a single fixed point: $c e^x$ is a fixed point for each constant $c$. In the case of polynomials, after a finite number of iterations ($1+d$ for a polynomial of degree $d$) you end up at the fixed point $0$.