Duality of differential operators and algebraic de Rham cohomology

Given a smooth proper morphism $f\text{\hspace{0.17em}⁣}:X\to S$, we introduce a certain derived category where morphisms are permitted to be ${\mathcal{O}}_S$-linear differential operators. We then prove a generalisation of Serre duality that applies to two-term complexes of this type. We apply this to give a new proof of Poincaré duality for relative algebraic de Rham cohomology.