Editing Partial derivative

{{short description|Derivative of a function with multiple variables}}
{{Calculus}}

In [[mathematics]], a '''partial derivative''' of a [[function (mathematics)#MULTIVARIATE FUNCTION|function of several variables]] is its [[derivative]] with respect to one of those variables, with the others held constant (as opposed to the [[total derivative]], in which all variables are allowed to vary). Partial derivatives are used in [[vector calculus]] and [[differential geometry]].

The partial derivative of a function <math>f(x, y, \dots)</math> with respect to the variable <math>x</math> is variously denoted by
{{block indent | em = 1.2 | text = <math>f_x</math>, <math>f'_x</math>, <math>\partial_x f</math>, <math>\ D_xf</math>, <math>D_1f</math>, <math>\frac{\partial}{\partial x}f</math>, or <math>\frac{\partial f}{\partial x}</math>.}}
It can be thought of as the rate of change of the function in the <math>x</math>-direction.

Sometimes, for {{nowrap|<math>z=f(x, y, \ldots)</math>,}} the partial derivative of <math>z</math> with respect to <math>x</math> is denoted as <math>\tfrac{\partial z}{\partial x}.</math> Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in:

<math display="block">f'_x(x, y, \ldots), \frac{\partial f}{\partial x} (x, y, \ldots).</math>

The symbol used to denote partial derivatives is [[∂]]. One of the first known uses of this symbol in mathematics is by [[Marquis de Condorcet]] from 1770,<ref name="Cajori_History_V2">{{citation |last= Cajori |first= Florian |year= 1952 |title= A History of Mathematical Notations |at= 596 |edition = 3 |volume = 2  |publisher= The Open Court Publishing Company |url=https://archive.org/details/AHistoryOfMathematicalNotationVolII/page/n153/mode/2up}}</ref> who used it for [[partial difference equation|partial differences]]. The modern partial derivative notation was created by [[Adrien-Marie Legendre]] (1786), although he later abandoned it; [[Carl Gustav Jacob Jacobi]] reintroduced the symbol in 1841.<ref name="miller_earliest">{{citation |last= Miller |first= Jeff |date= n.d. |contribution= Earliest Uses of Symbols of Calculus |contribution-url= https://mathshistory.st-andrews.ac.uk/Miller/mathsym/calculus/ |access-date= 2023-06-15 |editor-last= O'Connor |editor-first= John J. |editor2-last= Robertson |editor2-first= Edmund F. |editor2-link= Edmund F. Robertson |title= [[MacTutor History of Mathematics archive]] |publisher= [[University of St Andrews]] |mode= cs1}}</ref>

==Definition==
Like ordinary derivatives, the partial derivative is defined as a [[limit of a function|limit]]. Let {{mvar|U}} be an [[open set|open subset]] of <math>\R^n</math> and <math>f:U\to\R</math> a function. The partial derivative of {{mvar|f}} at the point <math>\mathbf{a}=(a_1, \ldots, a_n) \in U</math> with respect to the {{mvar|i}}-th variable {{math|''x''<sub>''i''</sub>}} is defined as

<math display="block">\begin{align}
\frac{\partial }{\partial x_i }f(\mathbf{a}) & = \lim_{h \to 0} \frac{f(a_1, \ldots , a_{i-1}, a_i+h, a_{i+1}\, \ldots ,a_{n})\ -
f(a_1, \ldots, a_i, \dots ,a_n)}{h} \\ 
& = \lim_{h \to 0} \frac{f(\mathbf{a}+h\mathbf{e_i}) - f(\mathbf{a})}{h}\,.
\end{align}</math>

Where <math>\mathbf{e_i}</math> is the [[unit vector]] of {{mvar|i}}-th variable {{math|''x''<sub>''i''</sub>}}. Even if all partial derivatives <math>\partial f / \partial x_i(a)</math> exist at a given point {{mvar|a}}, the function need not be [[continuous function|continuous]] there. However, if all partial derivatives exist in a [[neighborhood (topology)|neighborhood]] of {{mvar|a}} and are continuous there, then {{mvar|f}} is [[total derivative|totally differentiable]] in that neighborhood and the total derivative is continuous. In this case, it is said that {{mvar|f}} is a {{math|''C''<sup>1</sup>}} function. This can be used to generalize for vector valued functions, {{nowrap|<math>f:U \to \R^m</math>,}} by carefully using a componentwise argument.

The partial derivative <math display="inline">\frac{\partial f}{\partial x}</math> can be seen as another function defined on {{mvar|U}} and can again be partially differentiated. If the direction of derivative is {{em|not}} repeated, it is called a '''''mixed partial derivative'''''. If all mixed second order partial derivatives are continuous at a point (or on a set), {{mvar|f}} is termed a {{math|''C''<sup>2</sup>}} function at that point (or on that set); in this case, the partial derivatives can be exchanged by [[Symmetry of second derivatives#Schwarz's theorem|Clairaut's theorem]]:

<math display="block">\frac{\partial^2f}{\partial x_i \partial x_j} = \frac{\partial^2f} {\partial x_j \partial x_i}.</math>

==Notation==
{{see|∂}}
For the following examples, let {{mvar|f}} be a function in {{mvar|x}}, {{mvar|y}}, and {{mvar|z}}.

First-order partial derivatives:

<math display="block">\frac{ \partial f}{ \partial x} = f'_x = \partial_x f.</math>

Second-order partial derivatives:

<math display="block">\frac{ \partial^2 f}{ \partial x^2} = f''_{xx} = \partial_{xx} f = \partial_x^2 f.</math>

Second-order [[mixed derivatives]]:

<math display="block">\frac{\partial^2 f}{\partial y \,\partial x} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = (f'_{x})'_{y} = f''_{xy} = \partial_{yx} f = \partial_y \partial_x f .</math>

Higher-order partial and mixed derivatives:

<math display="block">\frac{\partial^{i+j+k} f}{\partial x^i \partial y^j \partial z^k } = f^{(i, j, k)} = \partial_x^i \partial_y^j \partial_z^k f.</math>

When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as [[statistical mechanics]], the partial derivative of {{mvar|f}} with respect to {{mvar|x}}, holding {{mvar|y}} and {{mvar|z}} constant, is often expressed as

<math display="block">\left( \frac{\partial f}{\partial x} \right)_{y,z} .</math>

Conventionally, for clarity and simplicity of notation, the partial derivative ''function'' and the ''value'' of the function at a specific point are [[Abuse of notation|conflated]] by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like 

<math display="block">\frac{\partial f(x,y,z)}{\partial x}</math> 

is used for the function, while 

<math display="block">\frac{\partial f(u,v,w)}{\partial u}</math> 

might be used for the value of the function at the point {{nowrap|<math>(x,y,z)=(u,v,w)</math>.}} However, this convention breaks down when we want to evaluate the partial derivative at a point like {{nowrap|<math>(x,y,z)=(17, u+v, v^2)</math>.}} In such a case, evaluation of the function must be expressed in an unwieldy manner as 

<math display="block">\frac{\partial f(x,y,z)}{\partial x}(17, u+v, v^2)</math> 

or 

<math display="block">\left. \frac{\partial f(x,y,z)}{\partial x}\right |_{(x,y,z)=(17, u+v, v^2)}</math> 

in order to use the Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with <math>D_i</math> as the partial derivative symbol with respect to the {{mvar|i}}-th variable. For instance, one would write <math>D_1 f(17, u+v, v^2)</math> for the example described above, while the expression <math>D_1 f</math> represents the partial derivative ''function'' with respect to the first variable.<ref>{{Cite book |last= Spivak |first= M. |date= 1965 |title= Calculus on Manifolds |publisher= W. A. Benjamin |location= New York |pages= 44 |isbn= 9780805390216 |url=https://archive.org/details/SpivakM.CalculusOnManifoldsPerseus2006Reprint }}</ref>

For higher order partial derivatives, the partial derivative (function) of <math>D_i f</math> with respect to the {{mvar|j}}-th variable is denoted {{nowrap|<math>D_j(D_i f)=D_{i,j} f</math>.}} That is, {{nowrap|<math>D_j\circ D_i =D_{i,j}</math>,}} so that the variables are listed in the order in which the derivatives are taken, and thus, in reverse order of how the composition of operators is usually notated. Of course, [[Clairaut's theorem on equality of mixed partials|Clairaut's theorem]] implies that <math>D_{i,j}=D_{j,i}</math> as long as comparatively mild regularity conditions on {{mvar|f}} are satisfied.

==Gradient==
{{Main|Gradient}}
An important example of a function of several variables is the case of a [[scalar-valued function]] <math>f(x_1, \ldots, x_n)</math> on a domain in Euclidean space <math>\R^n</math> (e.g., on <math>\R^2</math> or {{nowrap|<math>\R^3</math>).}} In this case {{mvar|f}} has a partial derivative <math>\partial f/\partial x_j</math> with respect to each variable {{math|''x''<sub>''j''</sub>}}. At the point {{mvar|a}}, these partial derivatives define the vector

<math display="block">\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \ldots, \frac{\partial f}{\partial x_n}(a)\right).</math>

This vector is called the ''[[gradient]]'' of {{mvar|f}} at {{mvar|a}}. If {{mvar|f}} is differentiable at every point in some domain, then the gradient is a vector-valued function {{math|∇''f''}} which takes the point {{mvar|a}} to the vector {{math|∇''f''(''a'')}}. Consequently, the gradient produces a [[vector field]].

A common [[abuse of notation]] is to define the [[del operator]] ({{math|∇}}) as follows in three-dimensional [[Euclidean space]] <math>\R^3</math> with [[unit vectors]] {{nowrap|<math>\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}</math>:}}

<math display="block">\nabla = \left[{\frac{\partial}{\partial x}} \right] \hat{\mathbf{i}} + \left[{\frac{\partial}{\partial y}} \right] \hat{\mathbf{j}} + \left[{\frac{\partial}{\partial z}}\right] \hat{\mathbf{k}}</math>

Or, more generally, for {{mvar|n}}-dimensional Euclidean space <math>\R^n</math> with coordinates <math>x_1, \ldots, x_n</math> and unit vectors {{nowrap|<math>\hat{\mathbf{e}}_1, \ldots, \hat{\mathbf{e}}_n</math>:}}

<math display="block">\nabla = \sum_{j=1}^n \left[\frac{\partial}{\partial x_j} \right] \hat{\mathbf{e}}_j = \left[\frac{\partial}{\partial x_1} \right] \hat{\mathbf{e}}_1 + \left[\frac{\partial}{\partial x_2} \right] \hat{\mathbf{e}}_2 + \dots + \left[\frac{\partial}{\partial x_n} \right] \hat{\mathbf{e}}_n</math>

==Directional derivative==
{{excerpt|Directional derivative|Definition}}

== Example ==
Suppose that {{mvar|f}} is a function of more than one variable. For instance,

<math display="block">z = f(x,y) = x^2 + xy + y^2 .</math>

{{multiple image
 | align  = right
 | direction = vertical
 | width  = 250

 | image1 = Partial func eg.svg
 | caption1 = A graph of {{nowrap|1={{math|1=''z'' = ''x''<sup>2</sup> + ''xy'' + ''y''<sup>2</sup>}}}}. For the partial derivative at {{nowrap|(1, 1)}} that leaves {{mvar|y}} constant, the corresponding [[tangent]] line is parallel to the {{mvar|xz}}-plane.

 | image2 = X2+X+1.svg
 | caption2 = A slice of the graph above showing the function in the {{mvar|xz}}-plane at {{nowrap|1={{math|1=''y'' = 1}}}}. The two axes are shown here with different scales. The slope of the tangent line is 3.
}}

The [[graph of a function|graph]] of this function defines a [[Surface (topology)|surface]] in [[Euclidean space]]. To every point on this surface, there are an infinite number of [[tangent line]]s. Partial differentiation is the act of choosing one of these lines and finding its [[slope]]. Usually, the lines of most interest are those that are parallel to the {{mvar|xz}}-plane, and those that are parallel to the {{mvar|yz}}-plane (which result from holding either {{mvar|y}} or {{mvar|x}} constant, respectively).

To find the slope of the line tangent to the function at {{math|''P''(1, 1)}} and parallel to the {{mvar|xz}}-plane, we treat {{mvar|y}} as a constant. The graph and this plane are shown on the right. Below, we see how the function looks on the plane {{math|1=''y'' = 1}}. By finding the [[derivative]] of the equation while assuming that {{mvar|y}} is a constant, we find that the slope of {{mvar|f}} at the point {{math|(''x'', ''y'')}} is:

<math display="block">\frac{\partial z}{\partial x} = 2x+y.</math>

So at {{math|(1, 1)}}, by substitution, the slope is {{math|3}}. Therefore,

<math display="block">\frac{\partial z}{\partial x} = 3</math>

at the point {{math|(1, 1)}}. That is, the partial derivative of {{mvar|z}} with respect to {{mvar|x}} at {{math|(1, 1)}} is {{math|3}}, as shown in the graph.

The function {{mvar|f}} can be reinterpreted as a family of functions of one variable indexed by the other variables:

<math display="block">f(x,y) = f_y(x) = x^2 + xy + y^2.</math>

In other words, every value of {{mvar|y}} defines a function, denoted {{math|''f<sub>y</sub>''}}, which is a function of one variable {{mvar|x}}.<ref>This can also be expressed as the [[adjoint functors|adjointness]] between the [[product topology|product space]] and [[function space]] constructions.</ref> That is,

<math display="block">f_y(x) = x^2 + xy + y^2.</math>

In this section the subscript notation {{math|''f<sub>y</sub>''}} denotes a function contingent on a fixed value of {{mvar|y}}, and not a partial derivative.

Once a value of {{mvar|y}} is chosen, say {{mvar|a}}, then {{math|''f''(''x'',''y'')}} determines a function {{math|''f<sub>a</sub>''}} which traces a curve {{math|1=''x''<sup>2</sup> + ''ax'' + ''a''<sup>2</sup>}} on the {{mvar|xz}}-plane:

<math display="block">f_a(x) = x^2 + ax + a^2.</math>

In this expression, {{mvar|a}} is a {{em|constant}}, not a {{em|variable}}, so {{math|''f<sub>a</sub>''}} is a function of only one real variable, that being {{mvar|x}}. Consequently, the definition of the derivative for a function of one variable applies:

<math display="block">f_a'(x) = 2x + a.</math>

The above procedure can be performed for any choice of {{mvar|a}}. Assembling the derivatives together into a function gives a function which describes the variation of {{mvar|f}} in the {{mvar|x}} direction:

<math display="block">\frac{\partial f}{\partial x}(x,y) = 2x + y.</math>

This is the partial derivative of {{mvar|f}} with respect to {{mvar|x}}. Here '{{mvar|∂}}' is a rounded 'd' called the ''[[partial derivative symbol]]''; to distinguish it from the letter 'd', '{{mvar|∂}}' is sometimes pronounced "partial".

==Higher order partial derivatives==

Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the function <math>f(x, y, ...)</math> the "own" second partial derivative with respect to {{mvar|x}} is simply the partial derivative of the partial derivative (both with respect to {{mvar|x}}):<ref>{{cite book |last= Chiang |first= Alpha C. |date= 1984 |title= Fundamental Methods of Mathematical Economics |publisher= McGraw-Hill |edition= 3rd |author-link= Alpha Chiang }}</ref>{{rp|316–318}}

<math display="block">\frac{\partial ^2 f}{\partial x^2} \equiv \partial \frac{{\partial f / \partial x}}{{\partial x}} \equiv \frac{{\partial f_x }}{{\partial x }} \equiv f_{xx}.</math>

The cross partial derivative with respect to {{mvar|x}} and {{mvar|y}} is obtained by taking the partial derivative of {{mvar|f}} with respect to {{mvar|x}}, and then taking the partial derivative of the result with respect to {{mvar|y}}, to obtain

<math display="block">\frac{\partial ^2 f}{\partial y\, \partial x} \equiv \partial \frac{\partial f / \partial x}{\partial y} \equiv \frac{\partial f_x}{\partial y} \equiv f_{xy}.</math>

[[Schwarz theorem|Schwarz's theorem]] states that if the second derivatives are continuous, the expression for the cross partial derivative is unaffected by which variable the partial derivative is taken with respect to first and which is taken second. That is,

<math display="block">\frac {\partial ^2 f}{\partial x\, \partial y} = \frac{\partial ^2 f}{\partial y\, \partial x}</math>

or equivalently <math>f_{yx} = f_{xy}.</math>

Own and cross partial derivatives appear in the [[Hessian matrix]] which is used in the [[second order condition]]s in [[optimization]] problems.
The higher order partial derivatives can be obtained by successive differentiation

==Antiderivative analogue==
There is a concept for partial derivatives that is analogous to [[antiderivative]]s for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function.

Consider the example of 

<math display="block">\frac{\partial z}{\partial x} = 2x+y.</math>

The so-called partial integral can be taken with respect to {{mvar|x}} (treating {{mvar|y}} as constant, in a similar manner to partial differentiation):

<math display="block">z = \int \frac{\partial z}{\partial x} \,dx = x^2 + xy + g(y).</math>

Here, the [[constant of integration]] is no longer a constant, but instead a function of all the variables of the original function except {{mvar|x}}. The reason for this is that all the other variables are treated as constant when taking the partial derivative, so any function which does not involve {{mvar|x}} will disappear when taking the partial derivative, and we have to account for this when we take the antiderivative. The most general way to represent this is to have the constant represent an unknown function of all the other variables.

Thus the set of functions {{nowrap|<math>x^2 + xy + g(y)</math>,}} where {{mvar|g}} is any one-argument function, represents the entire set of functions in variables {{math|''x'', ''y''}} that could have produced the {{mvar|x}}-partial derivative {{nowrap|<math>2x + y</math>.}}

If all the partial derivatives of a function are known (for example, with the [[gradient]]), then the antiderivatives can be matched via the above process to reconstruct the original function up to a constant. Unlike in the single-variable case, however, not every set of functions can be the set of all (first) partial derivatives of a single function. In other words, not every vector field is [[Conservative vector field|conservative]].

==Applications==

===Geometry===
[[Image:Cone 3d.png|thumb|The volume of a cone depends on height and radius]]
The [[volume]] {{mvar|V}} of a [[cone (geometry)|cone]] depends on the cone's [[height]] {{mvar|h}} and its [[radius]] {{mvar|r}} according to the formula

<math display="block">V(r, h) = \frac{\pi r^2 h}{3}.</math>

The partial derivative of {{mvar|V}} with respect to {{mvar|r}} is

<math display="block">\frac{ \partial V}{\partial r} = \frac{ 2 \pi r h}{3},</math>

which represents the rate with which a cone's volume changes if its radius is varied and its height is kept constant. The partial derivative with respect to {{mvar|h}} equals {{nowrap|<math display="inline">\frac{1}{3}\pi r^2</math>,}} which represents the rate with which the volume changes if its height is varied and its radius is kept constant.

By contrast, the [[total derivative|''total'' derivative]] of {{mvar|V}} with respect to {{mvar|r}} and {{mvar|h}} are respectively

<math display="block">\begin{align}
  \frac{dV}{dr} &= \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r} + \overbrace{\frac{\pi r^2}{3}}^\frac{ \partial V}{\partial h}\frac{dh}{dr}\,, \\
  \frac{dV}{dh} &= \overbrace{\frac{\pi r^2}{3}}^\frac{\partial V}{\partial h} + \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r}\frac{dr}{dh}\,.
\end{align}</math>

The difference between the total and partial derivative is the elimination of indirect dependencies between variables in partial derivatives.

If (for some arbitrary reason) the cone's proportions have to stay the same, and the height and radius are in a fixed ratio {{mvar|k}},

<math display="block">k = \frac{h}{r} = \frac{dh}{dr}.</math>

This gives the total derivative with respect to {{mvar|r}},

<math display="block">\frac{dV}{dr} = \frac{2 \pi r h}{3} + \frac{\pi r^2}{3}k\,,</math>

which simplifies to

<math display="block">\frac{dV}{dr} = k \pi r^2,</math>

Similarly, the total derivative with respect to {{mvar|h}} is

<math display="block">\frac{dV}{dh} = \pi r^2.</math>

The total derivative with respect to {{em|both}} {{mvar|r}} and {{mvar|h}} of the volume intended as scalar function of these two variables is given by the [[gradient]] vector 

<math display="block">\nabla V = \left(\frac{\partial V}{\partial r},\frac{\partial V}{\partial h}\right) = \left(\frac{2}{3}\pi rh, \frac{1}{3}\pi r^2\right).</math>

===Optimization===

Partial derivatives appear in any calculus-based [[optimization]] problem with more than one choice variable. For example, in [[economics]] a firm may wish to maximize [[profit (economics)|profit]] {{math|π(''x'', ''y'')}} with respect to the choice of the quantities {{mvar|x}} and {{mvar|y}} of two different types of output. The [[first order condition]]s for this optimization are {{math|1= π<sub>''x''</sub> = 0 = π<sub>''y''</sub>}}. Since both partial derivatives {{math|π<sub>''x''</sub>}} and {{math|π<sub>''y''</sub>}} will generally themselves be functions of both arguments {{mvar|x}} and {{mvar|y}}, these two first order conditions form a [[System of equations|system of two equations in two unknowns]].

===Thermodynamics, quantum mechanics and mathematical physics===

Partial derivatives appear in thermodynamic equations like [[Gibbs-Duhem equation]], in quantum mechanics as in [[Schrödinger equation|Schrödinger wave equation]], as well as in other equations from [[mathematical physics]]. The variables being held constant in partial derivatives here can be ratios of simple variables like [[mole fraction]]s {{math|''x<sub>i</sub>''}} in the following example involving the Gibbs energies in a ternary mixture system:

<math display="block">\bar{G_2}= G + (1-x_2) \left(\frac{{\partial G}}{{\partial x_2}}\right)_{\frac{x_1}{x_3}} </math>

Express [[mole fraction]]s of a component as functions of other components' mole fraction and binary mole ratios:

<math display="inline">\begin{align}
  x_1 &= \frac{1-x_2}{1+\frac{x_3}{x_1}} \\
  x_3 &= \frac{1-x_2}{1+\frac{x_1}{x_3}}
\end{align}</math>

Differential quotients can be formed at constant ratios like those above:

<math display="block">\begin{align}
  \left(\frac{\partial x_1}{\partial x_2}\right)_{\frac{x_1}{x_3}} &= - \frac{x_1}{1-x_2} \\
  \left(\frac{\partial x_3}{\partial x_2}\right)_{\frac{x_1}{x_3}} &= - \frac{x_3}{1-x_2}
\end{align}</math>

Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems:

<math display="block">\begin{align}
  X &= \frac{x_3}{x_1 + x_3} \\
  Y &= \frac{x_3}{x_2 + x_3} \\
  Z &= \frac{x_2}{x_1 + x_2}
\end{align}</math>

which can be used for solving [[partial differential equation]]s like:

<math display="block">\left(\frac{\partial \mu_2}{\partial n_1}\right)_{n_2, n_3} = \left(\frac{\partial \mu_1}{\partial n_2}\right)_{n_1, n_3}</math>

This equality can be rearranged to have differential quotient of mole fractions on one side.

===Image resizing===

Partial derivatives are key to target-aware image resizing algorithms. Widely known as [[seam carving]], these algorithms require each [[pixel]] in an image to be assigned a numerical 'energy' to describe their dissimilarity against orthogonal adjacent pixels. The [[algorithm]] then progressively removes rows or columns with the lowest energy. The formula established to determine a pixel's energy (magnitude of [[gradient]] at a pixel) depends heavily on the constructs of partial derivatives.

===Economics===

Partial derivatives play a prominent role in [[economics]], in which most functions describing economic behaviour posit that the behaviour depends on more than one variable. For example, a societal [[consumption function]] may describe the amount spent on consumer goods as depending on both income and wealth; the [[marginal propensity to consume]] is then the partial derivative of the consumption function with respect to income.

==See also==
{{div col|colwidth=27em}}
*[[d'Alembert operator]]
*[[Chain rule]]
*[[Curl (mathematics)]]
*[[Divergence]]
*[[Exterior derivative]]
*[[Iterated integral]]
*[[Jacobian matrix and determinant]]
*[[Laplace operator]]
*[[Multivariable calculus]]
*[[Symmetry of second derivatives]]
*[[Triple product rule]], also known as the cyclic chain rule.
{{div col end}}

== Notes ==
{{reflist}}

== External links ==
* {{Springer |title = Partial derivative |id = p/p071620 }}
* [http://mathworld.wolfram.com/PartialDerivative.html Partial Derivatives] at [[MathWorld]]

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{{Calculus topics}}

[[Category:Multivariable calculus]]
[[Category:Differential operators]]