Fourier transform of $x\left ( t \right )= t\cdot e^{-t^{2}}$

Question

Knowing that the Fourier transform of the function $y\left ( t \right )= e^{-t^{2}}$ is equal to $\sqrt{\pi}e^{-\frac{\omega ^{2}}{4}}$ I then proceeded using the the derivative rule $$\mathcal {F}\left [ te^{-t^{2}} \right ]=-i\sqrt{\pi} \frac{\omega }{2}e^{-\frac{\omega ^{2}}{4}}$$ However my answer is not listed as a possible solution. I tried tackling this problem by using definition, but there appeared some nasty integrals.