If f is a periodic function, then

f' and f'' are also periodic

Let f(x) be a periodic function with period T. Then, f(x + T) = f(x) for all x

Now,

$f'(x)=\lim\limits_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

∴ $f'(x+T)=\lim\limits_{h \rightarrow 0} \frac{f(x+T+h)-f(x+T)}{h}$

$\Rightarrow f'(x+T)=\lim\limits_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ = f'(x) for all x

Similarly, we have

f''(x + T) = f''(x) for all x

Hence, f' and f'' are also periodic functions with the same period.