= Solution
Under regularity and integrability conditions permitting the limits to pass through the integral, apply the difference quotient directly in time:
$$
\begin{aligned}
\mathcal F_x[u_t](k,t)&=\lim_{h\to0}\int_{\mathbb R}\frac{u(x,t+h)-u(x,t)}h e^{-ikx}dx\\
&=\lim_{h\to0}\frac{\widetilde u(k,t+h)-\widetilde u(k,t)}h=\partial_t\widetilde u(k,t).
\end{aligned}
$$
For the spatial derivative, translate the integration variable in the first term:
$$
\begin{aligned}
\mathcal F_x[u_x](k,t)&=\lim_{h\to0}\int_{\mathbb R}\frac{u(x+h,t)-u(x,t)}h e^{-ikx}dx\\
&=\lim_{h\to0}\frac{e^{ikh}-1}{h}\widetilde u(k,t)=ik\widetilde u(k,t).
\end{aligned}
$$
Consequently
$$
\boxed{\mathcal F_x[u_t]=\partial_t\widetilde u,\qquad\mathcal F_x[u_x]=ik\widetilde u.}
$$
The second calculation is the <Translation property of the Fourier transform> applied to a difference quotient. These are the <Fourier transform of a derivative> identities. The stated decay removes boundary terms in the alternative integration-by-parts derivation; rigorous interchange of limits also requires suitable control, for example integrable derivatives. The formulas extend to <tempered distributions>.
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