Hasse–Schmidt derivations, divided powers and differential smoothness

Let k be a commutative ring, A a commutative k-algebra and D the filtered ring of k-linear differential operators of A. We prove that: (1) The graded ring grD admits a canonical embedding θ into the graded dual of the symmetric algebra of the module ΩA/k of differentials of A over k, which has a canonical divided power structure. (2) There is a canonical morphism θ from the divided power algebra of the module of k-linear Hasse-Schmidt integrable derivations of A to grD. (3) Morphisms θ and θ fit into a canonical commutative diagram.