For some example functions that grow more slowly than $e^n$, like $2n^3$, or any polynomial, their derivative grows more slowly than them (for $2n^3$ it goes down to $6n^2$).
For $e^n$, its derivative exactly equals it.
For some example functions that grow more quickly than $e^n$ (i.e. $f$ such that $f = \Omega(e^n)$), like $e^{2n}$, their derivative grows more quickly than them (for $e^{2n}$ it goes up to $2e^{2n}$).
So I'm wondering, are the examples I gave indicative of a generally applicable statement about $e^n$ being the cutoff between functions whose derivative grows more quickly than them and functions who derivative grows more slowly than them, or did I just choose some well-behaving functions which make it look like that but there are actually counterexamples to this supposed pattern?
EDIT: A comment brought up the counterexample $sin(e^x)$. So let's add the constraint of the functions being monotonic. Perhaps that's unnecessarily strict, but I think it's a good starting place which is still true to the spirit of my original curiosity.