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{{Short description|Fundamental construction of differential calculus}}
{{about|the term as used in mathematics|other uses|derivative (disambiguation)}}
{{Calculus |Differential}}
In [[mathematics]], the [[derivative]] is a fundamental construction of [[differential calculus]] and admits many possible generalizations within the fields of [[mathematical analysis]], [[combinatorics]], [[algebra]], [[geometry]], etc.

== Fréchet derivative ==

The [[Fréchet derivative]] defines the derivative for general [[normed vector space]]s <math>V, W</math>. Briefly, a function <math>f : U \to W</math>, where <math>U</math> is an open subset of <math>V</math>, is called ''Fréchet differentiable'' at <math>x \in U</math> if there exists a [[bounded linear operator]] <math>A:V\to W</math> such that
<math display=block>\lim_{\|h\| \to 0} \frac{\| f(x + h) - f(x) - Ah\|_W}{\|h\|_V} = 0.</math>

Functions are defined as being differentiable in some open [[neighbourhood (mathematics)|neighbourhood]]  of <math>x</math>, rather than at individual points, as not doing so tends to lead to many [[Pathological (mathematics)|pathological]] [[counterexamples]].

The Fréchet derivative is quite similar to the formula for the [[derivative]] found in elementary one-variable calculus, <math display=block>\lim_{h \to 0}\frac{f(x+h) - f(x)}{h} = A,</math>  and simply moves ''A'' to the left hand side. However, the Fréchet derivative ''A'' denotes the function <math>t \mapsto f'(x) \cdot t</math>.

In [[multivariable calculus]], in the context of differential equations defined by a vector valued function '''R'''<sup>''n''</sup> to '''R'''<sup>''m''</sup>, the Fréchet derivative ''A'' is a [[linear operator]] on '''R''' considered as a vector space over itself, and corresponds to the ''best linear approximation'' of a function. If such an operator exists, then it is unique, and can be represented by an ''m'' by ''n'' [[matrix (mathematics)|matrix]] known as the [[Jacobian matrix and determinant|Jacobian matrix]] J<sub>''x''</sub>(ƒ) of the mapping ƒ at point ''x''. Each entry of this matrix represents a [[partial derivative]], specifying the rate of change of one range coordinate with respect to a change in a domain coordinate. Of course, the Jacobian
matrix of the composition ''g<sub>°</sub>f'' is a product of corresponding Jacobian matrices:    
J<sub>''x''</sub>(''g<sub>°</sub>f'') =J<sub>ƒ(''x'')</sub>(''g'')J<sub>''x''</sub>(ƒ). This is a higher-dimensional statement of the [[chain rule]].

For real valued functions from '''R'''<sup>''n''</sup> to '''R''' ([[scalar field]]s), the Fréchet derivative corresponds to a [[vector field]] called the [[total derivative]]. This can be interpreted as the [[gradient]] but it is more natural to use the [[exterior derivative]].

The [[convective derivative]] takes into account changes due to time dependence and motion through space along a vector field, and is a special case of the total derivative.

For [[vector-valued functions]] from '''R''' to '''R'''<sup>''n''</sup> (i.e., [[parametric curve]]s), the Fréchet derivative corresponds to taking the derivative of each component separately. The resulting derivative can be mapped to a vector. This is useful, for example, if the vector-valued function is the position vector of a particle through time, then the derivative is the velocity vector of the particle through time.

In [[complex analysis]], the central objects of study are [[holomorphic function]]s, which are complex-valued functions on the [[complex numbers]] where the Fréchet derivative exists.

In [[geometric calculus]], the [[Geometric_calculus#Differentiation|geometric derivative]] satisfies a weaker form of the Leibniz (product) rule. It specializes the Fréchet derivative to the objects of geometric algebra. Geometric calculus is a powerful formalism that has been shown to encompass the similar frameworks of differential forms and differential geometry.<ref>[[David Hestenes]], Garrett Sobczyk: Clifford Algebra to Geometric Calculus, a Unified Language for mathematics and Physics (Dordrecht/Boston:G.Reidel Publ.Co., 1984, {{ISBN|90-277-2561-6}}</ref>

== Exterior derivative and Lie derivative ==

On the [[exterior algebra]] of [[differential forms]] over a [[smooth manifold]], the [[exterior derivative]] is the unique linear map which satisfies a [[Graded Leibniz rule|graded version of the Leibniz law]] and squares to zero.  It is a grade 1 derivation on the exterior algebra. In '''R'''<sup>3</sup>, the [[gradient]], [[curl (mathematics)|curl]], and [[divergence]] are special cases of the exterior derivative. An intuitive interpretation of the gradient is that it points "up": in other words, it points in the direction of fastest increase of the function.  It can be used to calculate [[directional derivative]]s of [[Scalar (mathematics)|scalar]] functions or normal directions. Divergence gives a measure of how much "source" or "sink" near a point there is. It can be used to calculate [[flux]] by [[divergence theorem]]. Curl measures how much "[[rotation]]" a vector field has near a point.

The [[Lie derivative]] is the rate of change of a vector or tensor field along the flow of another vector field.  On vector fields, it is an example of a [[Lie bracket]] (vector fields form the [[Lie algebra]] of the [[diffeomorphism group]] of the manifold).  It is a grade 0 derivation on the algebra.

Together with the [[interior product]] (a degree -1 derivation on the exterior algebra defined by contraction with a vector field), the exterior derivative and the Lie derivative form a [[Lie superalgebra]].

== Differential topology ==

In [[differential topology]], a [[vector field]] may be defined as a derivation on the ring of [[smooth function]]s on a [[manifold]], and a [[tangent vector]] may be defined as a derivation at a point.  This allows the abstraction of the notion of a [[directional derivative]] of a scalar function to general manifolds.  For manifolds that are [[subset]]s of '''R'''<sup>''n''</sup>, this tangent vector will agree with the [[directional derivative]].

The [[pushforward (differential)|differential or pushforward]] of a map between manifolds is the induced map between tangent spaces of those maps.  It abstracts the [[Jacobian matrix]].

== Covariant derivative ==

In [[differential geometry]], the [[covariant derivative]] makes a choice for taking directional derivatives of vector fields along [[curve]]s.  This extends the directional derivative of scalar functions to sections of [[vector bundle]]s or [[principal bundle]]s.  In [[Riemannian geometry]], the existence of a metric chooses a unique preferred [[Torsion tensor|torsion]]-free covariant derivative, known as the [[Levi-Civita connection]].  See also [[gauge covariant derivative]] for a treatment oriented to physics.

The [[exterior covariant derivative]] extends the exterior derivative to vector valued forms.

== Weak derivatives ==
Given a function <math>u:\R^n\to\R</math> which is [[Locally integrable function|locally integrable]], but not necessarily classically differentiable, a [[weak derivative]] may be defined by means of [[integration by parts]]. First define test functions, which are infinitely differentiable and compactly supported functions <math>\varphi \in C^{\infty}_c\left(\R^n\right)</math>, and [[Multi-index notation|multi-indices]], which are length <math>n</math> lists of integers <math>\alpha = (\alpha_1, \dots, \alpha_n)</math> with <math display="inline">|\alpha| := \sum_1^n \alpha_i</math>. Applied to test functions, <math display="inline">D^\alpha \varphi := \frac{\partial^{|\alpha|} \varphi}{\partial x_1^{\alpha_1} \dotsm x_n^{\alpha_n}}</math>. Then the <math display="inline">\alpha^{\text{th}} </math> weak derivative of <math>u</math> exists if there is a function <math>v:\R^n\to\R</math> such that for ''all'' test functions <math>\varphi</math>, we have 

: <math>\int_{\R^n} u\ D^{\alpha} \!\varphi\ dx = (-1)^{|\alpha|}\int_{\R^n} v\ \varphi\ dx</math>

If such a function exists, then <math>D^{\alpha} u := v </math>, which is unique [[almost everywhere]]. This definition coincides with the classical derivative for functions <math>u \in C^{|\alpha|}\left(\R^n\right) </math>, and can be extended to a type of generalized functions called [[Distribution (mathematics)|distributions]], the dual space of test functions. Weak derivatives are particularly useful in the study of partial differential equations, and within parts of functional analysis.

== Higher-order and fractional derivatives ==

In the real numbers one can iterate the differentiation process, that is, apply derivatives more than once, obtaining derivatives of second and higher order. Higher derivatives can also be defined for functions of several variables, studied in [[multivariable calculus]]. In this case, instead of repeatedly applying the derivative, one repeatedly applies [[partial derivative]]s with respect to different variables. For example, the second order partial derivatives of a scalar function of ''n'' variables can be organized into an ''n'' by ''n'' matrix, the [[Hessian matrix]].  One of the subtle points is that the higher derivatives are not intrinsically defined, and depend on the choice of the coordinates in a complicated fashion (in particular, the Hessian matrix of a function is not a [[tensor]]). Nevertheless, higher derivatives have important applications to analysis of [[maxima and minima|local extrema]] of a function at its [[critical point (mathematics)|critical points]]. For an advanced application of this analysis to topology of [[manifold]]s, see  [[Morse theory]].

In addition to ''n''&thinsp;th derivatives for any natural number ''n'', there are various ways to define derivatives of fractional or negative orders, which are studied in [[fractional calculus]]. The &minus;1 order derivative corresponds to the integral, whence the term [[differintegral]].

== Quaternionic derivatives ==
In [[quaternionic analysis]], derivatives can be defined in a similar way to real and complex functions. Since the [[Quaternion|quaternions]] <math>\mathbb{H}</math> are not commutative, the limit of the difference quotient yields two different derivatives: A left derivative
:<math>\lim_{h \to 0} \left[h^{-1} \left(f(a+h) - f(a) \right) \right]</math>
and a right derivative
:<math>\lim_{h \to 0}\left[\left(f(a+h) - f(a) \right) h^{-1} \right].</math>

The existence of these limits are very restrictive conditions. For example, if <math>f:\mathbb{H} \to \mathbb{H}</math> has left-derivatives at every point on an open connected set <math>U \subset \mathbb{H}</math>, then <math>f(q) = a + qb</math> for <math>a,b \in \mathbb{H}</math>.

==Difference operator, q-analogues and time scales==

* The [[q-derivative]]  of a function is defined by the formula <math display="block"> D_q f(x)=\frac{f(qx)-f(x)}{(q-1)x}.</math> For ''x'' nonzero, if ''f'' is a differentiable function of ''x'' then in the limit as {{math|''q'' → 1}} we obtain the ordinary derivative, thus the ''q''-derivative may be viewed as its [[q-deformation]]. A large body of results from ordinary differential calculus, such as [[binomial formula]] and [[Taylor expansion]], have natural ''q''-analogues that were discovered in the 19th century, but remained relatively obscure for a big part of the 20th century, outside of the theory of [[special functions]]. The progress of [[combinatorics]] and the discovery of [[quantum group]]s have changed the situation dramatically, and the popularity of ''q''-analogues is on the rise.
* The [[difference operator]] of [[difference equations]] is another discrete analog of the standard derivative. <math display="block">\Delta f(x)=f(x+1)-f(x)</math>
* The q-derivative, the difference operator and the standard derivative can all be viewed as the same thing on different [[time scale calculus|time scales]]. For example, taking <math>\varepsilon = (q-1)x </math>, we may have <math display="block"> \frac{f(qx)-f(x)}{(q-1)x} = \frac{f(x+\varepsilon)-f(x)}{\varepsilon}.</math> The q-derivative is a special case of the [[Wolfgang Hahn| Hahn]] difference,<ref>{{cite journal |last=Hahn |first=Wolfgang |authorlink=Wolfgang Hahn |title=Über Orthogonalpolynome, die q-Differenzengleichungen genügen |year=1949 |journal=[[Mathematische Nachrichten]] |issn=0025-584X |volume=2 |issue=1–2 |pages=4–34 |doi=10.1002/mana.19490020103 |mr=0030647}}</ref> <math display="block"> \frac{f(qx+\omega)-f(x)}{qx+\omega-x}.</math>The Hahn difference is not only a generalization of the q-derivative but also an extension of the forward difference.
* Also note that the q-derivative is nothing but a special case of the familiar derivative. Take <math> z = qx </math>. Then we have, <math display="block">\lim_{z \to x}\frac{f(z) - f(x)}{z - x} = \lim_{q \to 1}\frac{f(qx) - f(x)}{qx - x} = \lim_{q \to 1}\frac{f(qx)-f(x)}{(q-1)x}.</math>

== Derivatives in algebra ==
In algebra, generalizations of the derivative can be obtained by imposing the [[product rule|Leibniz rule of differentiation]] in an algebraic structure, such as a [[ring (mathematics)|ring]] or a [[Lie algebra]].

=== Derivations ===

A [[derivation (abstract algebra)|derivation]] is a linear map on a  ring or [[algebra over a field|algebra]] which satisfies the Leibniz law (the product rule).  Higher derivatives and [[algebraic differential equation|algebraic differential operators]] can also be defined. They are studied in a purely algebraic setting in [[differential Galois theory]] and the theory of [[D-module]]s, but also turn up in many other areas, where they often agree with less algebraic definitions of derivatives.

For example, the [[differential algebra|formal derivative]] of a [[polynomial]] over a commutative ring ''R'' is defined by
:<math>\left(a_d x^d + a_{d-1}x^{d-1} + \cdots + a_1x + a_0\right)' = da_d x^{d-1} + (d - 1)a_{d-1}x^{d-2} + \cdots + a_1.</math>
The mapping <math>f\mapsto f'</math> is then a derivation on the [[polynomial ring]] ''R''[''X''].  This definition can be extended to [[rational function]]s as well.

The notion of derivation applies to noncommutative as well as commutative rings, and even to non-associative algebraic structures, such as Lie algebras.

=== Derivative of a type ===
In [[type theory]], many [[abstract data type]]s can be described as the [[universal algebra|algebra]] generated by a transformation that maps structures based on the type back into the type. For example, the type T of [[binary tree]]s containing values of type A can be represented as the algebra generated by the transformation 1+A&times;T<sup>2</sup>→T. The "1" represents the construction of an empty tree, and the second term represents the construction of a tree from a value and two subtrees. The "+" indicates that a tree can be constructed either way.

The derivative of such a type is the type that describes the context of a particular substructure with respect to its next outer containing structure. Put another way, it is the type representing the "difference" between the two. In the tree example, the derivative is a type that describes the information needed, given a particular subtree, to construct its parent tree. This information is a tuple that contains a binary indicator of whether the child is on the left or right, the value at the parent, and the sibling subtree. This type can be represented as 2&times;A&times;T, which looks very much like the derivative of the transformation that generated the tree type.

This concept of a derivative of a type has practical applications, such as the [[zipper (data structure)|zipper]] technique used in [[functional programming language]]s.

== Differential operators ==

A [[differential operator]] combines several derivatives, possibly of different orders, in one algebraic expression. This is especially useful in considering ordinary [[linear differential equation]]s with constant coefficients. For example, if ''f''(''x'') is a twice differentiable function of one variable, the differential equation <math>f'' + 2f' - 3f = 4x - 1</math> may be rewritten in the form  <math>L(f)=4x-1</math>, where 
<math display=block> L=\frac{d^2}{dx^2}+2\frac{d}{dx}-3</math>
is a ''second order linear constant coefficient differential operator'' acting on functions of ''x''. The key idea here is that we consider a particular [[linear combination]] of zeroth, first and second order derivatives "all at once". This allows us to think of the set of solutions of this differential equation as a "generalized antiderivative" of its right hand side 4''x''&nbsp;−&nbsp;1, by analogy with ordinary [[Integral|integration]], and formally write 
<math display="block">f(x)=L^{-1}(4x-1).</math>

Combining derivatives of different variables results in a notion of a [[partial differential operator]]. The [[linear operator]] which assigns to each function its derivative is an example of a differential operator on a [[function space]]. By means of the [[Fourier transform]], [[pseudo-differential operator]]s can be defined which allow for fractional calculus.

Some of these operators are so important that they have their own names:

* The [[Laplace operator]] or Laplacian on '''R'''<sup>3</sup> is a second-order partial differential operator {{math|Δ}} given by the  [[divergence]] of the [[gradient]] of a scalar function of three variables, or explicitly as <math display="block"> \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}. </math> Analogous operators can be defined for functions of any number of variables. 
* The [[d'Alembertian]] or wave operator is similar to the Laplacian, but acts on functions of four variables. Its definition uses the indefinite [[metric tensor]] of [[Minkowski space]], instead of the [[Euclidean space|Euclidean]] [[dot product]] of '''R'''<sup>3</sup>: <math display="block"> \square = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}. </math>
* The [[Schwarzian derivative]] is a non-linear differential operator which describes how a complex function is approximated by a [[fractional-linear map]], in much the same way that a normal derivative describes how a function is approximated by a linear map.
* The [[Wirtinger derivatives]] are a set of differential operators that permit the construction of a differential calculus for complex functions that is entirely analogous to the ordinary differential calculus for functions of real variables.

== Other generalizations ==

In [[functional analysis]], the [[functional derivative]] defines the derivative with respect to a function of a functional on a space of functions.  This is an extension of the directional derivative to an infinite [[dimension]]al vector space. An important case is the variational derivative in the [[calculus of variations]]. 

The [[subderivative]] and [[subgradient]] are generalizations of the derivative to [[convex function]]s used in convex analysis.

In [[commutative algebra]], [[Kähler differential]]s are universal derivations of a [[commutative ring]] or [[module (algebra)|module]]. They can be used to define an analogue of exterior derivative from differential geometry that applies to arbitrary [[algebraic varieties]], instead of just smooth manifolds.

In [[p-adic analysis]], the usual definition of derivative is not quite strong enough, and one requires [[strictly differentiable|strict differentiability]] instead.

The [[Gateaux derivative]] extends the Fréchet derivative to [[locally convex]] [[topological vector space]]s.  Fréchet differentiability is a strictly stronger condition than Gateaux differentiability, even in finite dimensions.  Between the two extremes is the [[quasi-derivative]].

In [[measure theory]], the [[Radon–Nikodym derivative]] generalizes the [[Jacobian matrix and determinant|Jacobian]], used for changing variables, to measures. It expresses one measure μ in terms of another measure ν (under certain conditions).

The [[H-derivative|''H''-derivative]] is a notion of derivative in the study of [[abstract Wiener space]]s and the [[Malliavin calculus]].  It is used in the study of [[stochastic processes]].

Laplacians and differential equations using the Laplacian can be [[analysis on fractals|defined on fractals]]. There is no completely satisfactory analog of the first-order derivative or gradient.<ref>[https://www.ams.org/notices/199910/fea-strichartz.pdf Analysis on Fractals], Robert S. Strichartz - Article in Notices of the AMS</ref>

{{Anchor|Analogues_of_derivatives_in_fields_of_positive_characteristic}}
The '''Carlitz derivative''' is an operation similar to usual differentiation but with the usual context of real or complex numbers changed to [[local fields]] of positive [[Characteristic_(algebra)|characteristic]] in the form of [[formal Laurent series]] with coefficients in some [[finite field]] F<sub>''q''</sub> (it is known that any local field of positive characteristic is isomorphic to a Laurent series field). Along with suitably defined analogs to the [[exponential function]], [[logarithms]] and others the derivative can be used to develop notions of smoothness, analycity, integration, Taylor series as well as a theory of differential equations.<ref>{{cite book |title=Analysis in Positive Characteristic |last=Kochubei |first= Anatoly N.|year=2009 |publisher= Cambridge University Press |location= New York |isbn= 978-0-521-50977-0}}</ref>

It may be possible to combine two or more of the above different notions of extension or abstraction of the original derivative.  For example, in [[Finsler geometry]], one studies spaces which look [[locally]] like [[Banach space]]s.  Thus one might want a derivative with some of the features of a [[functional derivative]] and the [[covariant derivative]].

Multiplicative calculus replaces addition with multiplication, and hence rather than dealing with the limit of a ratio of differences, it deals with the limit of an exponentiation of ratios. This allows the development of the geometric derivative and  bigeometric derivative. Moreover, just like the classical differential operator has a discrete analog, the difference operator, there are also [[List of derivatives and integrals in alternative calculi|discrete analogs of these multiplicative derivatives]].

==See also==

* {{annotated link|Arithmetic derivative}}
* {{annotated link|Automatic differentiation}}
* {{annotated link|Brzozowski derivative}}
* {{annotated link|Dini derivative}}
* {{annotated link|Fractal derivative}}
* {{annotated link|Hasse derivative}}
* {{annotated link|Logarithmic derivative}}
* {{annotated link|Logarithmic differentiation}}
* {{annotated link|Non-classical analysis}}
* {{annotated link|Numerical differentiation}}
* {{annotated link|Pincherle derivative}}
* {{annotated link|q-derivative}}
* {{annotated link|Semi-differentiability}}
* {{annotated link|Symmetric derivative}}
* {{annotated link|Topological derivative}}

==Notes==

{{reflist}}

{{Analysis in topological vector spaces}}

[[Category:Generalizations of the derivative| ]]