We regarded $xe^{-\pi y^2/2}$ and $\delta'(x)e^{-y^2/2}$ as distributions on $\mathcal{S}(\mathbb{R}^2)$, then find the Fourier transform of them.
My attempt
$<F[xe^{-\pi y^2/2}],\phi>=<xe^{-\pi y^2/2},F[\phi]>$
$=-<e^{-\pi y^2/2},F[\partial_1\phi]>$
$=<F[e^{-\pi y^2/2}],\partial_1\phi>$
Then the problem is to figure out the Fourier transform of $e^{-\pi y^2/2}$.