= Solution
<Weak convergence of distributions> means $\langle T_n,\varphi\rangle\to\langle T,\varphi\rangle$ for every <test function> $\varphi\in C_c^\infty(\Omega)$. The <distributional derivative> is defined by
$$
\boxed{\langle\partial_{x_j}T,\varphi\rangle=-\langle T,\partial_{x_j}\varphi\rangle.}
$$
It is linear and continuous: differentiation preserves compact support and bounds each test-function seminorm by one of higher derivative order, so composing it with the continuous functional $T$ again gives a <distribution>. Finally, for every fixed test function,
$$
\langle\partial_{x_j}T_n,\varphi\rangle
=-\langle T_n,\partial_{x_j}\varphi\rangle
\longrightarrow-\langle T,\partial_{x_j}\varphi\rangle
=\langle\partial_{x_j}T,\varphi\rangle.
$$
Thus distributional differentiation preserves the stated convergence.
Back to article page
