Derivative of Periodic Real Function

Theorem
Let $f: \mathbb F \to \mathbb F$ be a function, where $\mathbb F \in \left\{{\R, \C}\right\}$.

Let $f$ be differentiable.

If $f$ is periodic with period $L$, then its derivative is also periodic with period $L$.

Proof
Let $f$ be differentiable and periodic with period $L$.

Then:

The result follows from the definition of periodic.

Also see

 * Primitive of Periodic Function