in mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. specifically, given an algebra a over a ring or a field k, a k-derivation is a k-linear map d:&nbsp;a&nbsp;&rarr;&nbsp;a that satisfies leibniz's law: : d(ab) = (da)b + a(db). more generally, if m is an a-module, a k-linear map d:a&rarr;m which satisfies the leibniz law is also called a derivation. the collection of all k-derivations of a to itself is denoted by derk(a). the collection of k-derivations of a into an a-module m is denoted by derk(a,m). 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. the pincherle derivative is an example of a derivation in abstract algebra. if the algebra a is noncommutative, then the commutator with respect to an element of the algebra a defines a linear endomorphism of a to itself, which is a derivation over k. an algebra a equipped with a distinguished derivation d forms a differential algebra, and is itself a significant object of study in areas such as differential galois theory. ==properties== the leibniz law itself has a number of immediate consequences. firstly, if x1,&nbsp;x2,&nbsp;â¦&nbsp;,x'n&nbsp;&isin;&nbsp;a, then it follows by mathematical induction that :d(x_1x_2\cdots x_n) = \sum_i x_1\cdots x_{i-1}d(x_i)x_{i+1}\cdots x_n = \sum_i d(x_i)\prod_{j\neq i}x_j. \, in particular, if a is commutative and x1&nbsp;=&nbsp;x2&nbsp;=&nbsp;â¦&nbsp;=&nbsp;x'n, then this formula simplifies to the familiar power rule d(x'n) = nx'n&minus;1d(x). secondly, if a has a unit element 1, then d(1) = 0 since d(1) = d(1Â·1) = d(1)&nbsp;+&nbsp;d(1). moreover, because d is k-linear, it follows that &ldquo;the derivative of any constant function is zero&rdquo;; more precisely, for any x&nbsp;&isin;&nbsp;k, d(x)&nbsp;=&nbsp;d(xÂ·1)&nbsp;=&nbsp;xÂ·d(1)&nbsp;=&nbsp;0. if k&nbsp;&sub;&nbsp;k is a subring, and a is a k-algebra, then there is an inclusion :\operatorname{der}_k(a,m)\subset \operatorname{der}_k(a,m),\, since any k-deri
Differential_algebra