Derivatives and differentials are usually expressed in terms of limits in the sense of analysis. However it became clear in about the last half century that much of the knowledge on usual differential calculus can be inferred from using just algebraic properties of differentials and derivatives, most notably the Leibniz rule for differentiating products (); an alternative synthetic formalism also appeared which did not use limiting procedures as well.
A derivation of an associative algebra is simply a linear endomorphism satisfying the Leibniz rule. Then for example the tangent vector fields on a smooth manifold are obtained as derivations of the algebra of -functions on the manifold. The differential of a map is a linearized approximation. This is clear in various non-classical analytic setups, for example for maps between Banach spaces and for differentiable manifolds.
This linearization idea has been obtained at the level of sheaves of -modules by Grothendieck for algebraic varieties a the view toward the differential calculus for varieties in prime characteristics. It is interesting that he related differential calculus to resolutions of the diagonal, where he considered the sheaves of modules supported on infinitesimal neighborhoods of diagonal. Indeed, to define a derivative in analysis, one needs to start with consideration of differences of values of a function at points which are close to one to another, and , and this means that means that the pair is close to the diagonal of the cartesian square . In numerical analysis?, various approximation schemas for higher order differential operators are involved which obviously live around higher diagonals in ( times). One of the products of that thinking is Grothendieck’s notion of a regular differential operator. This led later to the creation of the theory of D-modules which are sheaves of modules over the sheaf of rings of regular differential operators over a scheme, or a complex analytic manifold.
We plan in the nLab to cover many aspects of the interaction between geometry and differential calculi of various sorts including synthetic differential geometry and algebraic counterparts of notions from differential calculi. It is hard to say, however, where homological algebra belongs: the differential in the sense of homological algebra is rather a notion which can be systematized into the more general subject of homotopical algebra, but in some cases it is related to analogues of exterior differentiation for the de Rham complex of differential forms (say on manifolds). But there is also an analogue of the Taylor series for functors in some homotopical contexts (say Goodwillie calculus).
One should also point out that an elaborate schema for differential calculus in noncommutative geometry has been proposed by Tsygan in terms of algebras with higher brackets. In one version, a differential calculus is given there by a Gerstenhaber algebra and a Batalin-Vilkovisky module over it.
See derivation, regular differential operator, differential form, differential bimodule, universal differential envelope, differential forms in synthetic differential geometry, connection, connection for a differential graded algebra, D-module, Fox derivative, crystal, Fermat theory…