Example of a function that's not periodic but its derivative is periodic?

Question

Show by example a function where $\frac{df}{dx}$ is periodic but $f$ is not periodic.

The book I'm reading has only shown me the derivatives of sine, cosine, tangent, secant, cosecant, and cotangent. As far as my knowledge goes, all these functions are periodic, and all their derivatives are periodic. I'm unable to think of an example where the function $f$ is not periodic but its derivative is periodic.

I considered how $\cot x$ is not defined for all odd multiples of $\frac{\pi}{2}$ but $-\csc^2x$ (its derivative) is defined for these values. But I know that that doesn't make the cotangent function non-periodic by any means.

Any help would be greatly appreciated.

Thanks.

Answer 1

If the integral of $\frac{df}{dx}$ over one period is non-zero, $f$ will not be periodic.

Answer 2

Pick your favorite derivable periodic function $g$ and consider $f : x \mapsto x + g(x)$.
$f$ is not periodic.
$f'(x) = 1 + g'(x)$. Since $g'$ is periodic, $f'$ is periodic, too.