Example: for function $$f(x)=x^{3}(1-x)^{3}=\sum f_{s}\exp(2\pi isx)$$ Fourier series of its fourth derivative are different from derivative of its Fourier series $$f^{(4)}(x)=-360x^{2}+360x-72=\sum g_{s}\exp(2\pi i s x)$$ with $g_{s}\neq(2\pi s)^{4}f_{s}$
Related question: When $$\sum\left|f_{s}\right|^{2}j^{2p}<\infty$$ is equivalent to $f\in C^{(p)}[0,1]$? (for periodic f)