Differentiation in time FT
Nobody as talked much about this one so i thought i should give it a try. We want to prove that the F.T. of $ \frac{d}{dt}x(t) $ is $ j\omega X(\omega) \,\ $
Proof: Differentiating the inverse Fourier Transform $ X(\omega) \,\ $ with respect to t we get.
$ \frac{d}{dt} [\frac{1}{2pi} \int_{-\infty}^{\infty} X(\omega)e^{j\omega t}d\omega ] $
$ = \frac{1}{2pi} \int_{-\infty}^{\infty} X(\omega) \frac{d}{dt} e^{j\omega t}d\omega $
$ = \frac{1}{2pi} \int_{-\infty}^{\infty} [j\omega X(\omega)] e^{j\omega t} d\omega $
$ = F^{-1}[j\omega X(\omega)] \,\ $
