The simplest example of this is $e^x$ which we could say has period 1 (it is its own derivative). $e^{-x}$ would have period 2.
Using similar constructions, I can get a function that has a derivative of period $n$ by doing $e^{x\cdot (1)^{1/n}}$
Is this the only way to get periodic derivatives?
Note: I am treating sin and cos as special cases of this when $n=4$
Is there a proof to this effect?