  We show for an affine variety $X$, the derived category of quasi-coherent
$D$-modules is equivalent to the category of DG modules over an explicit DG
algebra, whose zeroth cohomology is the ring of Grothendieck differential
operators $\operatorname{Diff}(X)$. When the variety is cuspidal, we show that
this is just the usual ring $\operatorname{Diff}(X)$, and the equivalence is
the abelian equivalence constructed by Ben-Zvi and Nevins. We compute the
cohomology algebra and its natural modules in the hypersurface, curve and
isolated quotient singularity cases. We identify cases where a $D$-module is
realised as an ordinary module (in degree 0) over $\operatorname{Diff}(X)$ and
where it is not.
