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Can anyone point me to a derivation of $x^n f(x)$? I know that the answer is $(i)^n$ times the $n$-th derivative of the transform of $f(x)$, but I've searched for a derivation and can't find it.

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    @GerryMyerson Sorry, I was not aware that it would be problematic. I will scale down the editing.2012-11-22

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It suffices to prove the case with $n=1$ and then apply induction.

Consider applying $d/ds$ to both sides of the definition (below) and then multiplying both sides by $i$.

$ F(s)=\mathcal{F}\{f(x)\}(s)=\int_{-\infty}^\infty f(x)e^{-isx}dx$

(We must have sufficient regularity on $f$ for this derivation to be valid.)