I wanted to prouve that : $$\mathcal{F}\{\frac{dx(t)}{dt}\} = 2i\pi f \mathcal{F}\{x(t)\}$$ I started by the applying the definition of Fourier Transform and I integrated by parts:
$\int_{-\infty}^{+\infty}\frac{dx(t)}{dt} e^{-2i\pi ft} dt$ = $\Big [x(t)e^{-2i\pi ft} \Big]_{-\infty}^{+\infty}$ + $ 2i\pi f\int_{-\infty}^{+\infty}x(t) e^{-2i\pi ft} dt$
but I faced a problem in calculating $\Big [x(t)e^{-2i\pi ft} \Big]_{-\infty}^{+\infty}$ I couldn't calculate the limit of this complex number
can I get some help with explanation? much appreciated