Using the quotient rule, we obtain that the notion of differential operator on a localization is well-defined:$$\mathcal{D}_A(B_f) \cong \mathcal{D}_A(B)_f.$$Here, $B$ is a commutative $A$-algebra, $\mathcal{D}_A(B)$ denotes the ring of differential operators of $B$ over $A$, and $B_f$ is $B$ localized about the multiplicative set generated by $f$, where $f \in B$.
Is there a good way to intuit/picture this geometrically?