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 $\text{gr}D$ admits a canonical embedding $\theta$ into the graded dual of the symmetric algebra of the module ${\mathrm{\Omega }}_{A\mathrm{/}k}$ of differentials of $A$ over $k$, which has a canonical divided power structure. (2) There is a canonical morphism $\vartheta$ from the divided power algebra of the module of $k$-linear Hasse-Schmidt integrable derivations of $A$ to $\text{gr}D$. (3) Morphisms $\theta$ and $\vartheta$ fit into a canonical commutative diagram.