= Solution
A sufficient assumption is that $f$ is absolutely continuous, $f,f'\in L^1(\mathbb R)$, and $f(t)\to0$ at both infinities. <Integration by parts> in the <Fourier transform> gives
$$
\widetilde{f'}(\omega)
=\left[f(t)e^{-i\omega t}\right]_{-\infty}^{\infty}
+i\omega\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt
=\boxed{i\omega\widetilde f(\omega)}.
$$
The decay assumption removes the boundary term, and integrability makes both transforms ordinary convergent integrals. The same <derivative> identity extends to <tempered distributions>.
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