  By reading a standard formula for the ring of Grothendieck differential
operators in a derived way, we construct a derived (sheaf of) ring of
Grothendieck differential operators for Noetherian schemes $X$ separated and
finite-type over a base $S$, when the map $X \to S$ is finite tor-amplitude.
Using this ring of differential operators, we (re-)develop the theory of
$D$-modules from scratch and show an equivalence of categories between
$D$-modules using our definition and crystals over the infinitesimal site.
