Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, of class $\mathcal{C}^{1}$. We assume that $f$ and its derivative $f^{\prime}$ are integrable on $\mathbb{R}$. Show that, for any real $\xi$, $\mathcal{F}\left(f^{\prime}\right)(\xi) = 2\mathrm{i}\pi\xi \mathcal{F}(f)(\xi)$.