Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D : A → A that satisfies Leibniz's law: More generally, if M is an A-bimodule, a K-linear map D : A → M that satisfies the Leibniz law is also called a derivation.
Derivations occur in many different contexts in diverse areas of mathematics.
The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on Rn.
The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold.
Examples of anti-derivations include the exterior derivative and the interior product acting on differential forms.
Graded derivations of superalgebras (i.e. Z2-graded algebras) are often called superderivations.
Hasse–Schmidt derivations are K-algebra homomorphisms Composing further with the map that sends a formal power series