Lemma 10.133.2. Let R \to S be a ring map. Let L, M, N be S-modules. If D : L \to M and D' : M \to N are differential operators of order k and k', then D' \circ D is a differential operator of order k + k'.
Proof. Let g \in S. Then the map which sends x \in L to
D'(D(gx)) - gD'(D(x)) = D'(D(gx)) - D'(gD(x)) + D'(gD(x)) - gD'(D(x))
is a sum of two compositions of differential operators of lower order. Hence the lemma follows by induction on k + k'. \square
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