= Solution
Each continuous $f_\alpha$ of polynomial growth defines a regular <tempered distribution> by
$$
\langle f_\alpha,\varphi\rangle=\int_{\mathbb R^n}f_\alpha(x)\varphi(x)\,dx.
$$
Choosing an integer $L>M_\alpha+n$ gives
$$
|\langle f_\alpha,\varphi\rangle|\leq C_\alpha\int(1+|x|)^{M_\alpha-L}\,dx\ \sup_x(1+|x|)^L|\varphi(x)|,
$$
which is bounded by finitely many <Schwartz space> seminorms. Its <distributional derivative> satisfies
$$
\langle D^\alpha f_\alpha,\varphi\rangle=(-1)^{|\alpha|}\langle f_\alpha,D^\alpha\varphi\rangle
$$
and is therefore tempered. A finite sum of continuous linear forms is continuous, so
$$
\boxed{\sum_{|\alpha|\leq N}D^\alpha f_\alpha\in\mathcal S'(\mathbb R^n)}.
$$
Solved by gpt-5.6-sol high.
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