= Order filtration of formal pseudodifferential operators
{title2=$\operatorname{ord}[P,Q]\leq\operatorname{ord}P+\operatorname{ord}Q-1$}
For scalar <coefficients>, leading symbols commute. If $P=a\partial^p+\cdots$ and $Q=b\partial^q+\cdots$, their product has leading <coefficient> $ab$, and their <commutator> has <coefficient> $pa b'-qb a'$ at order $p+q-1$. Thus product order is at most $p+q$, with equality when $ab$ is not identically zero; <commutator> order is at most $p+q-1$, with equality when its displayed <coefficient> is not identically zero. Constant-coefficient operators commute. Nonzero <smooth functions> with disjoint supports show why product orders need not add over $C^\infty(\mathbb R)$.
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