Is $f(x)=e^x$ the only solution to $f(f'(x))=f'(f(x))$?

Question

Is $f(x)=e^x$ the only solution to $f(f'(x))=f'(f(x))$?

In particular I'm interested in the qualitative properties of the such solutions.

Answer 1

As you can see from the comments there are plenty of real-valued functions satisfying your equation. Here are the examples found so far (posting this as CW so that the question has an answer):

1. $f(x) = 0$ and $f(x) = x$ (Thomas Andrews)
2. $f(x) = n\left(\frac{x}{n}\right)^n$ for any $n\in\mathbb{R}_+\text{\\}\{0\}$ (mickep)
3. $f(x) = ke^{x}$ for all $k\in\mathbb{R}$ (Ned)
4. $f(x) = a(x+1-a)$ for all $a\in\mathbb{R}$