Which derivatives are eventually periodic?
I have noticed that is $a_{n}=f^{(n)}(x)$, the sequence $a_{n}$ becomes eventually periodic for a multitude of $f(x)$.
If $f(x)$ was a polynomial, and $\operatorname{deg}(f(x))=n$, note that $f^{(n)}(x)=C$ if $C$ is a constant. This implies that $f^{(n+i)}(x)=0$ for every $i$ which is a natural number.
If $f(x)=e^x$, note that $f(x)=f'(x)$. This implies that $f^{(n)}(x)=e^x$ for every natural number $n$.
If $f(x)=\sin(x)$, note that $f'(x)=\cos(x), f''(x)=-\sin(x), f'''(x)=-\cos(x), f''''(x)=\sin(x)$.
This implies that $f^{(4n)}(x)=f(x)$ for every natural number $n$.
In a similar way, if $f(x)=\cos(x)$, $f^{(4n)}(x)=f(x)$ for every natural number $n$.
These appear to be the only functions whose derivatives become eventually periodic.
What are other functions whose derivatives become eventually periodic? What is known about them? Any help would be appreciated.