< Derivative of a periodic function.

Suppose $f$ is a periodic function with period $T$, i.e. 
\[f(x+T)=f(x)     \text{for all}    x.\]
Then 
\[f'(x+T)=\lim_{\epsilon \rightarrow 0} \frac{f(x+T+\epsilon)-f(x+T)}{\epsilon}\]
\[=\lim_{\epsilon \rightarrow 0} \frac{f(x+\epsilon)-f(x)}{\epsilon}=f'(x)\]
for all $x$. Hence
\[f'(x+T)=f'(x)    \text{for all}   x.\]
Hence $f'$ is a periodic function.