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Adjunctions and comonads in differential algebra

It is known that the construction of the ring of fractions S^A of a commutative ring A by a multiplicative subset S of A can be extended to the differential case. This means that for a given differential ring (A, d), the differential ring of fractions of (A, d) by S is constructed simply by defining a derivation operator on S~ X A in terms of the derivation operator d on A. We seek to explain in the categorical setting of adjunctions and comonads the reasons for which this and other constructions can be extended to the differential case. A natural product of this investigation is the construction of the differential affine scheme of a differential ring.