= Solution
The <distributional derivative> of $u$ is defined by
$$
\boxed{\langle u',\phi\rangle=-\langle u,\phi'\rangle\quad(\phi\in\mathcal D(a,b)).}
$$
It is a <distribution> because $\phi\mapsto\phi'$ is a continuous map of the <space of test functions> into itself. If $f\in C^1(a,b)$ represents a <regular distribution>, <integration by parts> gives
$$
\langle f',\phi\rangle=-\int_a^b f\phi'=\int_a^b f'\phi.
$$
The <boundary term> vanishes because $\phi$ has <compact support>. Thus the <distributional derivative> agrees with the ordinary <derivative>.
For $\operatorname{supp}\phi\subseteq K$, the assumed <order of a distribution> estimate gives
$$
|\langle u',\phi\rangle|\leq C_K\max_{0\leq j\leq m}\|\phi^{(j+1)}\|_\infty
\leq C_K\max_{0\leq j\leq m+1}\|\phi^{(j)}\|_\infty.
$$
Consequently $\boxed{\operatorname{ord}(u')\leq m+1}$.
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