Consider the complex valued function $f(t)$ where $t \in [0, 2\pi)$.
Under what conditions are $f(t)$ and $f'(t)$ orthogonal to each other? I'm defining orthogonal here to be
$$ \int_0^{2\pi} \overline{f(t)} f'(t) dt = 0$$
If $f(t)$ is real valued then this old question suggests a method of proof involving a Fourier decomposition, but I don't fully follow the proof and it's not clear to me if it's applicable to complex functions.