Let $\varphi \in \mathcal{S}(\mathbb R^n)$ and $\xi \in \mathbb R^n$, $\xi_l$ its $l$th coordinate. We can use that $x^{\alpha}\varphi$,$D^{\alpha}\varphi \in \mathcal{S}(\mathbb R^n)$, where $\alpha \in \mathbb N^n_0$ (the usual multiindex notation). How do I see that the following holds:
$\xi_l (\mathcal{F}\varphi)(\xi) =$$ i \over (2\pi)^{-n/2}$$\int_{\mathbb R^n}$$\partial \over \partial x_l$$(e^{-i\langle x, \xi \rangle})\varphi(x) dx$
without using that $\xi^{\alpha}(\mathcal F \varphi)(\xi) = (-i)^{|\alpha|} \mathcal F (D^{\alpha}\varphi)(\xi)$?
To see that, write out the definition of the Fourier transform, move the $\xi_l$ inside the integral, and compare $\xi_le^{-ix\cdot\xi}$ with $\partial_{x_l}e^{-ix\cdot\xi}$. Can you complete the puzzle with these hints? Let me know if some step needs elaboration.