= Solution
Use the angular-frequency <Fourier transform> convention
$$
\boxed{\widehat f(s)=\int_{-\infty}^{\infty}f(x)e^{-isx}\,dx.}
$$
For $f,f'\in L^1(\mathbb R)$ with the usual local absolute-continuity hypothesis, the <Fourier transform of a derivative> is
$$
\boxed{\widehat{f'}(s)=is\widehat f(s).}
$$
For example, this applies when $f$ is $C^1$ and both displayed functions are integrable. These assumptions make integration by parts valid and the endpoint terms vanish.
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