## Calculus 8th Edition

If $f(x)$ is periodic , then there exist a number $k$ such that $f(x+k)=f(x)$. Differentiate both side with respect to $x$, we have $f'(x+k). (x+k)'=f'(x)$ But $(x+k)'=\frac{d}{dx}(x+k)=1$, therefore, $f'(x+k)=f'(x)$ Thus, $f'(x)$ is periodic (and its period is same as $f(x)$). Hence, the given statement is TRUE.