Fourier transform of function similar to a Riesz kernel

Question

I am trying to prove that the Fourier transform of $$\frac{x_1 x_2} {|x|^4}$$ in $\mathcal{R}^2$ (in the sense of distributions) is a bounded multiplier given by $\frac{\xi_1 \xi_2}{|\xi|^2}$ but am having trouble with the quadratic term in $x_i$, since this is not as easy to handle as $\frac{x_1}{|x|^3}$ on the sphere. Can anyone help? Thank you.

Answer 1

$$\frac{x_1x_2}{|x|^4}=-\frac12\frac\partial{\partial x_2}\frac{x_1}{|x|^2}.$$