By noting that $\int_{-\infty}^{+\infty} \frac{\partial f}{\partial t}(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x = \mathcal{F}\left(\frac{\partial f}{\partial t}(t, \cdot)\right)(\xi)$ and using question 7, show that, for any real $\xi$ and any real $t > 0$, $\frac{\partial \hat{f}}{\partial t}(t, \xi) = -4\pi^{2} \xi^{2} \hat{f}(t, \xi)$.