Solution
= Solution
Differentiation in $x$ does not change the allowed growth order in $\theta$, while every $\theta$-derivative lowers it by one. More precisely, for further multi-indices $\gamma,\delta$,
$$
|D_x^\gamma D_\theta^\delta(D_x^\alpha D_\theta^\beta a)|
=|D_x^{\alpha+\gamma}D_\theta^{\beta+\delta}a|
\leq C(1+|\theta|)^{N-|\beta|-|\delta|}.
$$
Thus
$$
\boxed{D_x^\alpha D_\theta^\beta a
\in\operatorname{Sym}(X,\mathbb R^k;N-|\beta|)}.
$$