Editing Derivative (section)

== Generalizations ==
{{Main|Generalizations of the derivative}}

The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a [[linear approximation]] of the function at that point.
* An important generalization of the derivative concerns [[complex function]]s of [[Complex number|complex variable]]s, such as functions from (a domain in) the complex numbers <math>\C</math> to {{tmath|1= \C }}. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition.{{sfn|Roussos|2014|p=303}} If <math>\C</math> is identified with <math>\R^2</math> by writing a complex number <math>z</math> as {{tmath|1= x+iy }} then a differentiable function from <math>\C</math> to <math>\C</math> is certainly differentiable as a function from <math>\R^2</math> to <math>\R^2</math> (in the sense that its partial derivatives all exist), but the converse is not true in general: the complex derivative only exists if the real derivative is ''complex linear'' and this imposes relations between the partial derivatives called the [[Cauchy–Riemann equations]] – see [[holomorphic function]]s.{{sfn|Gbur|2011|pp=261–264}}
* Another generalization concerns functions between [[smooth manifold|differentiable or smooth manifolds]]. Intuitively speaking such a manifold <math>M</math> is a space that can be approximated near each point <math>x</math> by a vector space called its [[tangent space]]: the prototypical example is a [[smooth surface]] in {{tmath|1= \R^3 }}. The derivative (or differential) of a (differentiable) map <math>f:M\to N</math> between manifolds, at a point <math>x</math> in {{tmath|1= M }}, is then a [[linear map]] from the tangent space of <math>M</math> at <math>x</math> to the tangent space of <math>N</math> at {{tmath|1= f(x) }}. The derivative function becomes a map between the [[tangent bundle]]s of <math>M</math> and {{tmath|1= N }}. This definition is used in [[differential geometry]].{{sfn|Gray|Abbena|Salamon|2006|p=[https://books.google.com/books?id=owEj9TMYo7IC&pg=PA826 826]}}
* Differentiation can also be defined for maps between [[vector space]], such as [[Banach space]], in which those generalizations are the [[Gateaux derivative]] and the [[Fréchet derivative]].<ref>{{harvnb|Azegami|2020}}. See p. [https://books.google.com/books?id=e08AEAAAQBAJ&pg=PA209 209] for the Gateaux derivative, and p. [https://books.google.com/books?id=e08AEAAAQBAJ&pg=PA211 211] for the Fréchet derivative.</ref>
* One deficiency of the classical derivative is that very many functions are not differentiable. Nevertheless, there is a way of extending the notion of the derivative so that all [[continuous function|continuous]] functions and many other functions can be differentiated using a concept known as the [[weak derivative]]. The idea is to embed the continuous functions in a larger space called the space of [[distribution (mathematics)|distributions]] and only require that a function is differentiable "on average".{{sfn|Funaro|1992|pp=[https://books.google.com/books?id=CX4SXf3mdeUC&pg=PA84 84–85]}}
* Properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology; an example is [[differential algebra]]. Here, it consists of the derivation of some topics in abstract algebra, such as [[Ring (mathematics)|rings]], [[Ideal (ring theory)|ideals]], [[Field (mathematics)|field]], and so on.{{sfn|Kolchin|1973|pp=[https://books.google.com/books?id=yDCfhIjka-8C&pg=PA58 58], [https://books.google.com/books?id=yDCfhIjka-8C&pg=PA126 126]}}
* The discrete equivalent of differentiation is [[finite difference]]s. The study of differential calculus is unified with the calculus of finite differences in [[time scale calculus]].{{sfn|Georgiev|2018|p=[https://books.google.com/books?id=OJJVDwAAQBAJ&pg=PA8 8]}}
* The [[arithmetic derivative]] involves the function that is defined for the [[Integer|integers]] by the [[prime factorization]]. This is an analogy with the product rule.{{sfn|Barbeau|1961}}