I have heard that if $X$ is a smooth scheme over $k$, then we can calculate the sheaf of differential operators $\mathcal{D}_X$ by considering étale morphisms from an affine open set to $\mathbb{A}^n_k$, and these morphisms always exist since $X$ is smooth.
I was wondering if anyone could explain in more detail or show me a reference for how this étale morphism allows us to compute this?