= Solution
Differentiate the defining Fourier integral $n$ times with respect to $k$:
$$
\frac{d^n\widetilde f}{dk^n}=\int_{\mathbb R}(-ix)^nf(x)e^{-ikx}\,dx.
$$
For a nonnegative integer $n$, this gives
$$
\boxed{\mathcal F[x^nf(x)](k)=i^n\frac{d^n\widetilde f(k)}{dk^n}.}
$$
Integrability of $x^nf$ together with suitable lower moments justifies differentiation under the integral. The same multiplication-derivative identity holds more generally for <tempered distributions>.
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