Periodic Function

Differentiate both sides of $f(x + p) = f(x)$ and the chain rule gives $f'(x + p) = f'(x)$, so every derivative of a $2\pi$ periodic function is again $2\pi$ periodic. That is all periodicity buys. The four step cycle $\sin x \to \cos x \to -\sin x \to -\cos x \to \sin x$ comes from the specific rules $\frac{d}{dx}\sin x = \cos x$ and $\frac{d}{dx}\cos x = -\sin x$, not from repetition: $g(x) = \sin x + \sin 2x$ is $2\pi$ periodic, every derivative of it is too, and yet $g''''(x) = \sin x + 16\sin 2x$, which is nowhere near a return to $g$.

Accumulation is where the pattern can break. $\int_0^x \sin t \, dt = 1 - \cos x$ is still periodic, but $\int_0^x (1 + \sin t) \, dt = x + 1 - \cos x$ is not, because the integrand has a non zero average over one period and the running total drifts upward forever.