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{{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]], and [[geometry]].

== Derivatives in analysis ==
In real, complex, and functional analysis, derivatives are generalized to functions of several real or complex variables and functions between [[topological vector spaces]]. An important case is the [[functional derivative|variational derivative]] in the [[calculus of variations]]. Repeated application of differentiation leads to derivatives of higher order and differential operators.

=== Multivariable calculus ===
{{main|Fréchet derivative}}
The '''derivative''' is often met for the first time as an operation on a single real function of a single real variable. One of the simplest settings for generalizations is to vector valued functions of several variables (most often the domain forms a vector space as well). This is the field of [[multivariable calculus]].

In one-variable calculus, we say that a function <math>f: \R \to \R</math> is '''differentiable''' at a point ''x'' if the limit
:<math>\lim_{h \to 0}\frac{f(x+h) - f(x)}{h}</math>
exists. Its value is then the derivative ƒ'(''x''). A function is differentiable on an [[Interval (mathematics)|interval]] if it is differentiable at every point within the interval. Since the line <math>L(z) = f'(x)z - f'(x)x + f(x)</math> is tangent to the original function at the point <math>(x, f(x))</math>, the derivative can be seen as a way to find the ''best linear approximation'' of a function. If one ignores the constant term, setting <math>L(z) = f'(x)z</math>, ''L''(''z'') becomes an actual [[linear operator]] on '''R''' considered as a vector space over itself.

This motivates the following generalization to functions mapping '''R'''''m'' to '''R'''''n'': ƒ is differentiable at ''x'' if there exists a [[linear operator]] ''A''(''x'') (depending on ''x'') such that
:<math>\lim_{\|h\| \to 0}\frac{\|f(x+h) - f(x) - A(x)h\|}{\|h\|} = 0.</math>
Although this definition is perhaps not as explicit as the above, if such an operator exists, then it is unique, and in the one-dimensional case coincides with the original definition. (In this case the derivative is represented by a 1-by-1 matrix consisting of the sole entry ''f'''(''x'').) Note that, in general, we concern ourselves mostly with functions 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]].

An ''m'' by ''n'' [[matrix (mathematics)|matrix]], of the [[linear operator]] ''A''(''x'') is known as '''[[Jacobian matrix and determinant|Jacobian]]''' matrix '''J'''''x''(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°f'' is a product of corresponding Jacobian matrices:
'''J'''''x''(''g°f'') ='''J'''ƒ(''x'')(''g'')'''J'''''x''(ƒ). This is a higher-dimensional statement of the [[chain rule]].

For real valued functions from '''R'''''n'' to '''R''' ([[scalar field]]s), the total derivative can be interpreted as a [[vector field]] called the '''[[gradient]]'''. 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.

Several linear combinations of partial derivatives are especially useful in the context of differential equations defined by a vector valued function '''R'''''n'' to '''R'''''n''. The '''[[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]]. The '''[[curl (mathematics)|curl]]''' measures how much "[[rotation]]" a vector field has near a point.

For [[vector-valued functions]] from '''R''' to '''R'''''n'' (i.e., [[parametric curve]]s), one can take the derivative of each component separately. The resulting derivative is another vector valued function. 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.

The '''[[convective derivative]]''' takes into account changes due to time dependence and motion through space along vector field.

=== Convex analysis ===

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

=== Higher-order derivatives and differential operators ===
One can iterate the differentiation process, that is, apply derivatives more than once, obtaining derivatives of second and higher order. A more sophisticated idea is to combine several derivatives, possibly of different orders, in one algebraic expression, a [[differential operator]]. 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> &ensp;&ensp; where &ensp;&ensp; <math> 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'' − 1, by analogy with ordinary [[Integral|integration]], and formally write

: <math>f(x)=L^{-1}(4x-1).\, </math>

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]].

As in the case of functions of one variable, we can combine first and higher order partial derivatives to arrive at a notion of a [[partial differential operator]]. Some of these operators are so important that they have their own names:

*The [[Laplace operator]] or '''Laplacian''' on '''R'''3 is a second-order partial differential operator ''Δ'' given by the [[divergence]] of the [[gradient]] of a scalar function of three variables, or explicitly as

:: <math> \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'''''3'':

:: <math> \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>

=== Analysis on fractals ===
Laplacians and differential equations can be defined on [[analysis on fractals|fractals]].

=== Fractional derivatives ===
In addition to ''n''-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 -1 order derivative corresponds to the integral, whence the term '''[[differintegral]]'''.

=== Complex analysis ===

In [[complex analysis]], the central objects of study are [[holomorphic function]]s, which are complex-valued functions on the [[complex numbers]] satisfying a [[Fréchet derivative|suitably extended definition of differentiability]].

The '''[[Schwarzian derivative]]''' 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.

=== Functional analysis ===

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.

The '''[[Fréchet derivative]]''' allows the extension of the directional derivative to a general [[Banach space]]. The '''[[Gâteaux derivative]]''' extends the concept to [[locally convex]] [[topological vector space]]s. Fréchet differentiability is a strictly stronger condition than Gâteaux 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).

In the theory of [[abstract Wiener space]]s, the [[H-derivative|''H''-derivative]] defines a derivative in certain directions corresponding to the Cameron-Martin [[Hilbert space]].

The derivative also admits a generalization to the space of '''[[distribution (mathematics)|distributions]]''' on a space of functions using [[integration by parts]] against a suitably well-behaved subspace.

On a [[function space]], the [[linear operator]] which assigns to each function its derivative is an example of a '''[[differential operator]]'''. General differential operators include higher order derivatives. By means of the [[Fourier transform]], '''[[pseudo-differential operator]]s''' can be defined which allow for fractional calculus.

===Analogues of derivatives in fields of positive characteristic===
The [[Carlitz derivative]] is an operation similar to usual differentiation have been devised 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'''''q'' (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>

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

* The '''[[q-derivative]]''' of a function is defined by the formula

: <math> 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 ''q''&ensp;→ 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>\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]].

== 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 '''formal derivative''' of a [[polynomial]] over a commutative ring ''R'' is defined by
:<math>(a_dx^d + a_{d-1}x^{d-1} + \cdots+a_1x+a_0)' = da_dx^{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.

Also see [[Pincherle derivative]].

=== Commutative algebra ===

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.

=== Number theory ===

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

Also see [[arithmetic derivative]] and [[Hasse derivative]].

=== Type theory ===
Many [[abstract data type]]s in mathematics and [[computer science]] 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×T2→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×A×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.

== Derivatives in geometry ==
Main types of derivatives in geometry are Lie derivatives along a vector field, exterior differential, and covariant derivatives.

=== 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'''''n'', this tangent vector will agree with the directional derivative defined above.

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]].

On the [[exterior algebra]] of [[differential forms]] over a [[smooth manifold]], the '''[[exterior derivative]]''' is the unique linear map which satisfies a [[graded]] version of the Leibniz law and squares to zero. It is a grade 1 derivation on the exterior algebra.

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 geometry ===

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.

== Other generalizations ==

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]].

The study of [[stochastic processes]] requires a form of calculus known as the [[Malliavin calculus]]. One notion of derivative in this setting is the [[H-derivative|''H''-derivative]] of a function on an [[abstract Wiener space]].

== See also ==
*[[Arithmetic derivative]]
*[[Non-classical analysis]]

== Notes ==

{{reflist}}

[[Category:Differential calculus]]
[[Category:Generalizations of the derivative| ]]
[[Category:Real analysis]]
[[Category:Functional analysis]]
[[Category:Algebra]]
[[Category:Q-analogs]]
[[Category:Differential geometry]]

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