= Solution
A <phase function> is a real $C^\infty$ function on $X\times(\mathbb R^k\setminus\{0\})$, positively homogeneous of degree one in its frequency variable,
$$
\Phi(x,t\theta)=t\Phi(x,\theta)\quad(t>0),
$$
with nonvanishing total differential $d_{x,\theta}\Phi$. Vanishing of its frequency gradient alone is allowed; those critical directions are relevant to singularities.
With $\langle\theta\rangle=(1+|\theta|^2)^{1/2}$, the <symbol class> $\operatorname{Sym}(X;\mathbb R^k;N)=S^N_{1,0}$ consists of <smooth functions> $a$ such that, for every $K\Subset X$ and every pair of <multi-indices>,
$$
|\partial_x^\alpha\partial_\theta^\beta a(x,\theta)|\leq C_{K,\alpha,\beta}\langle\theta\rangle^{N-|\beta|},\qquad x\in K.
$$
There is no requirement that the constant be uniform in the <derivative> indices. We use $D=(1/i)\partial$; the factors of $i$ do not affect these estimates.
For $b=D_x^\alpha D_\theta^\beta a$, any further <derivative> satisfies
$$
|\partial_x^\mu\partial_\theta^\nu b|=|\partial_x^{\mu+\alpha}\partial_\theta^{\nu+\beta}a|\leq C\langle\theta\rangle^{N-|\beta|-|\nu|}.
$$
Thus \b[frequency <derivatives> lower symbol order and spatial <derivatives> preserve it]:
$$
\boxed{D_x^\alpha D_\theta^\beta a\in\operatorname{Sym}(X;\mathbb R^k;N-|\beta|).}
$$
This is the differentiation rule in <symbol calculus>.
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