In mathematics, differential refers to infinitesimal differences or to the derivatives of functions. The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology.
The notion of a differential motivates several concepts in differential geometry (and differential topology).
Differentials are also important in algebraic geometry, and there are several important notions.
The term differential has also been adopted in homological algebra and algebraic topology, because of the role the exterior derivative plays in de Rham cohomology: in a cochain complex , the maps (or coboundary operators) di are often called differentials. Dually, the boundary operators in a chain complex are sometimes called codifferentials.
The properties of the differential also motivate the algebraic notions of a derivation and a differential algebra.