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| {{other uses of|∂|∂}}
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| {{Calculus |Multivariable}}
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| In [[mathematics]], a '''partial derivative''' of a [[function (mathematics)|function]] of several variables is its [[derivative]] with respect to one of those variables, [[Ceteris paribus|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]].
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| The partial derivative of a function ''f'' with respect to the variable ''x'' is variously denoted by
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| : <math>f^\prime_x,\ f_x,\ \partial_x f, \frac{\partial}{\partial x}f, \text{ or } \frac{\partial f}{\partial x}</math>
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| The partial-derivative symbol is ∂. One of the first known uses of the symbol in mathematics is by [[Marquis de Condorcet]] from 1770, who used it for partial differences. The modern [[partial derivative]] notation is by [[Adrien-Marie Legendre]] (1786), though he later abandoned it; [[Carl Gustav Jacob Jacobi]] re-introduced the symbol in 1841.<ref name="jeff_earliest">{{cite web|url=http://jeff560.tripod.com/calculus.html|title=Earliest Uses of Symbols of Calculus|author=Jeff Miller|date=2009-06-14|work=Earliest Uses of Various Mathematical Symbols|accessdate=2009-02-20}}</ref>
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| ==Introduction==
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| Suppose that ''ƒ'' is a function of more than one variable. For instance,
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| :<math>z = f(x,y) = \,\! x^2 + xy + y^2.\,</math>
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| {{multiple image
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| | align = right
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| | direction = vertical
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| | width = 250
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| | image1 = Grafico 3d x2+xy+y2.png
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| | caption1 = A graph of {{nowrap|''z'' {{=}} ''x''<sup>2</sup> + ''xy'' + ''y''<sup>2</sup>}}. For the partial derivative at {{nowrap|(1, 1, 3)}} that leaves ''y'' constant, the corresponding [[tangent]] line is parallel to the ''xz''-plane.
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| | image2 = X2+x+1.png
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| | caption2 = A slice of the graph above showing the function in the ''xz''-plane at {{nowrap|''y''{{=}} 1}}
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| }}
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| The [[graph of a function|graph]] of this function defines a [[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 ''xz''-plane, and those that are parallel to the ''yz''-plane (which result from holding either y or x constant, respectively.)
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| To find the slope of the line tangent to the function at P{{nowrap|(1, 1, 3)}} that is parallel to the ''xz''-plane, the ''y'' variable is treated as constant. The graph and this plane are shown on the right. On the graph below it, we see the way the function looks on the plane {{nowrap|''y'' {{=}} 1}}. By finding the [[derivative]] of the equation while assuming that ''y'' is a constant, the slope of ''ƒ'' at the point {{nowrap|(''x'', ''y'', ''z'')}} is found to be:
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| : <math>\frac{\partial z}{\partial x} = 2x+y</math>
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| So at {{nowrap|(1, 1, 3)}}, by substitution, the slope is 3. Therefore
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| : <math>\frac{\partial z}{\partial x} = 3</math>
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| at the point {{nowrap|(1, 1, 3)}}. That is, the partial derivative of ''z'' with respect to ''x'' at {{nowrap|(1, 1, 3)}} is 3.
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| ==Definition==
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| === Basic definition ===
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| The function ''f'' can be reinterpreted as a family of functions of one variable indexed by the other variables:
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| :<math>f(x,y) = f_x(y) = \,\! x^2 + xy + y^2.\,</math>
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| In other words, every value of ''x'' defines a function, denoted ''f<sub>x</sub>'', which is a function of one variable.<ref>This can also be expressed as the [[adjoint functors|adjointness]] between the [[product topology|product space]] and [[function space]] constructions.</ref> That is,
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| :<math>f_x(y) = x^2 + xy + y^2.\,</math>
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| Once a value of ''x'' is chosen, say ''a'', then ''f''(''x'',''y'') determines a function ''f<sub>a</sub>'' which sends ''y'' to ''a''<sup>2</sup> + ''ay'' + ''y''<sup>2</sup>:
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| :<math>f_a(y) = a^2 + ay + y^2. \,</math>
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| In this expression, ''a'' is a ''constant'', not a ''variable'', so ''f<sub>a</sub>'' is a function of only one real variable, that being ''y''. Consequently, the definition of the derivative for a function of one variable applies:
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| :<math>f_a'(y) = a + 2y. \,</math>
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| The above procedure can be performed for any choice of ''a''. Assembling the derivatives together into a function gives a function which describes the variation of ''f'' in the ''y'' direction:
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| :<math>\frac{\partial f}{\partial y}(x,y) = x + 2y.\,</math>
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| This is the partial derivative of ''f'' with respect to ''y''. Here ∂ is a rounded ''d'' called the '''partial derivative symbol'''. To distinguish it from the letter ''d'', ∂ is sometimes pronounced "del" or "partial" instead of "dee".
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| In general, the '''partial derivative''' of a function ''f''(''x''<sub>1</sub>,...,''x''<sub>''n''</sub>) in the direction ''x<sub>i</sub>'' at the point (''a''<sub>1</sub>,...,''a<sub>n</sub>'') is defined to be:
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| :<math>\frac{\partial f}{\partial x_i}(a_1,\ldots,a_n) = \lim_{h \to 0}\frac{f(a_1,\ldots,a_i+h,\ldots,a_n) - f(a_1,\ldots, a_i, \dots,a_n)}{h}.</math>
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| In the above difference quotient, all the variables except ''x<sub>i</sub>'' are held fixed. That choice of fixed values determines a function of one variable <math>f_{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n}(x_i) = f(a_1,\ldots,a_{i-1},x_i,a_{i+1},\ldots,a_n)</math>, and by definition,
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| :<math>\frac{df_{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n}}{dx_i}(a_i) = \frac{\partial f}{\partial x_i}(a_1,\ldots,a_n).</math>
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| In other words, the different choices of ''a'' index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.
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| An important example of a function of several variables is the case of a [[scalar-valued function]] ''f''(''x''<sub>1</sub>,...''x''<sub>''n''</sub>) on a domain in Euclidean space '''R'''<sup>''n''</sup> (e.g., on '''R'''<sup>2</sup> or '''R'''<sup>3</sup>). In this case ''f'' has a partial derivative ∂''f''/∂''x''<sub>''j''</sub> with respect to each variable ''x''<sub>''j''</sub>. At the point ''a'', these partial derivatives define the vector
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| :<math>\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \ldots, \frac{\partial f}{\partial x_n}(a)\right).</math>
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| This vector is called the '''[[gradient]]''' of ''f'' at ''a''. If ''f'' is differentiable at every point in some domain, then the gradient is a vector-valued function ∇''f'' which takes the point ''a'' to the vector ∇''f''(''a''). Consequently, the gradient produces a [[vector field]].
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| A common [[abuse of notation]] is to define the [[del operator]] (∇) as follows in three-dimensional [[Euclidean space]] '''R'''<sup>3</sup> with [[unit vectors]] <math>\mathbf{\hat{i}}, \mathbf{\hat{j}}, \mathbf{\hat{k}}</math>:
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| :<math>\nabla = \bigg[{\frac{\partial}{\partial x}} \bigg] \mathbf{\hat{i}} + \bigg[{\frac{\partial}{\partial y}}\bigg] \mathbf{\hat{j}} + \bigg[{\frac{\partial}{\partial z}}\bigg] \mathbf{\hat{k}}</math>
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| Or, more generally, for ''n''-dimensional Euclidean space '''R'''<sup>''n''</sup> with coordinates (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>,...,x<sub>''n''</sub>) and unit vectors (<math>\mathbf{\hat{e}_1}, \mathbf{\hat{e}_2}, \mathbf{\hat{e}_3}, \dots , \mathbf{\hat{e}_n}</math>):
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| :<math>\nabla = \sum_{j=1}^n \bigg[{\frac{\partial}{\partial x_j}}\bigg] \mathbf{\hat{e}_j} = \bigg[{\frac{\partial}{\partial x_1}}\bigg] \mathbf{\hat{e}_1} + \bigg[{\frac{\partial}{\partial x_2}}\bigg] \mathbf{\hat{e}_2} + \bigg[{\frac{\partial}{\partial x_3}}\bigg] \mathbf{\hat{e}_3} + \dots + \bigg[{\frac{\partial}{\partial x_n}}\bigg] \mathbf{\hat{e}_n}</math>
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| ===Formal definition===
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| Like ordinary derivatives, the partial derivative is defined as a [[limit of a function|limit]]. Let ''U'' be an [[open set|open subset]] of '''R'''<sup>''n''</sup> and ''f'' : ''U'' → '''R''' a function. The partial derivative of ''f'' at the point '''''a''''' = (''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>) ∈ ''U'' with respect to the ''i''-th variable ''a''<sub>''i''</sub> is defined as
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| :<math>\frac{ \partial }{\partial a_i }f(\mathbf{a}) =
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| \lim_{h \rightarrow 0}{
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| f(a_1, \dots , a_{i-1}, a_i+h, a_{i+1}, \dots ,a_n) -
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| f(a_1, \dots, a_i, \dots ,a_n) \over h }
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| </math>
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| Even if all partial derivatives ∂''f''/∂''a''<sub>''i''</sub>(''a'') exist at a given point ''a'', the function need not be [[continuous function|continuous]] there. However, if all partial derivatives exist in a [[neighborhood (topology)|neighborhood]] of ''a'' and are continuous there, then ''f'' is [[total derivative|totally differentiable]] in that neighborhood and the total derivative is continuous. In this case, it is said that ''f'' is a C<sup>1</sup> function. This can be used to generalize for vector valued functions (''f'' : ''U'' → ''R'''<sup>''m''</sup>) by carefully using a componentwise argument.
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| The partial derivative <math>\frac{\partial f}{\partial x}</math> can be seen as another function defined on ''U'' and can again be partially differentiated. If all mixed second order partial derivatives are continuous at a point (or on a set), ''f'' is termed a 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#Clairaut.27s theorem|Clairaut's theorem]]:
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| :<math>\frac{\partial^2f}{\partial x_i\, \partial x_j} = \frac{\partial^2f} {\partial x_j\, \partial x_i}.</math>
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| ==Examples==
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| [[Image:Cone 3d.png|thumb|The volume of a cone depends on height and radius]]
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| The [[volume]] ''V'' of a [[cone (geometry)|cone]] depends on the cone's [[height]] ''h'' and its [[radius]] ''r'' according to the formula
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| :<math>V(r, h) = \frac{\pi r^2 h}{3}.</math>
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| The partial derivative of ''V'' with respect to ''r'' is
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| :<math>\frac{ \partial V}{\partial r} = \frac{ 2 \pi r h}{3},</math>
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| 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 ''h'' is
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| :<math>\frac{ \partial V}{\partial h} = \frac{\pi r^2}{3},</math>
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| which represents the rate with which the volume changes if its height is varied and its radius is kept constant.
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| By contrast, the [[total derivative|''total'' derivative]] of ''V'' with respect to ''r'' and ''h'' are respectively
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| :<math>\frac{\operatorname dV}{\operatorname 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{\operatorname d h}{\operatorname d r}</math>
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| and
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| :<math>\frac{\operatorname dV}{\operatorname 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{\operatorname d r}{\operatorname d h}</math>
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| The difference between the total and partial derivative is the elimination of indirect dependencies between variables in partial derivatives.
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| If (for some arbitrary reason) the cone's proportions have to stay the same, and the height and radius are in a fixed ratio ''k'',
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| :<math>k = \frac{h}{r} = \frac{\operatorname d h}{\operatorname d r}.</math>
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| This gives the total derivative with respect to ''r'':
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| :<math>\frac{\operatorname dV}{\operatorname dr} = \frac{2 \pi r h}{3} + \frac{\pi r^2}{3}k</math>
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| Which simplifies to:
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| :<math>\frac{\operatorname dV}{\operatorname dr} = k\pi r^2</math>
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| Similarly, the total derivative with respect to ''h'' is:
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| :<math>\frac{\operatorname dV}{\operatorname dh} = \pi r^2</math>
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| Equations involving an unknown function's partial derivatives are called [[partial differential equation]]s and are common in [[physics]], [[engineering]], and other [[science]]s and applied disciplines.
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| The total derivative with respect to ''both'' r and h is given by the [[Jacobian matrix and determinant|Jacobian matrix]], which here takes the form of the [[gradient]] vector <math>\nabla V =(\frac{\partial V}{\partial r},\frac{\partial V}{\partial h}) = (\frac{2}{3}\pi rh, \frac{1}{3}\pi r^2)</math>.
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| ==Notation==
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| For the following examples, let ''f'' be a function in ''x'', ''y'' and ''z''.
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| First-order partial derivatives:
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| :<math>\frac{ \partial f}{ \partial x} = f_x = \partial_x f.</math>
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| Second-order partial derivatives:
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| :<math>\frac{ \partial^2 f}{ \partial x^2} = f_{xx} = \partial_{xx} f.</math>
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| Second-order [[mixed derivatives]]:
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| :<math>\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.</math>
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| Higher-order partial and mixed derivatives:
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| :<math>\frac{ \partial^{i+j+k} f}{ \partial x^i\, \partial y^j\, \partial z^k } = f^{(i, j, k)}.</math>
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| When dealing with functions of multiple variables, some of these variables may be related to each other, and it may be necessary to specify explicitly which variables are being held constant. In fields such as [[statistical mechanics]], the partial derivative of ''f'' with respect to ''x'', holding ''y'' and ''z'' constant, is often expressed as
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| :<math>\left( \frac{\partial f}{\partial x} \right)_{y,z}.</math>
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| ==Antiderivative analogue==
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| 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.
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| Consider the example of <math>\frac{\partial z}{\partial x} = 2x+y</math>. The "partial" integral can be taken with respect to ''x'' (treating ''y'' as constant, in a similar manner to partial differentiation):
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| :<math>z = \int \frac{\partial z}{\partial x} \,dx = x^2 + xy + g(y)</math>
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| Here, the [[Constant of integration|"constant" of integration]] is no longer a constant, but instead a function of all the variables of the original function except ''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 <math>x</math> 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.
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| Thus the set of functions <math>x^2 + xy + g(y)</math>, where ''g'' is any one-argument function, represents the entire set of functions in variables ''x'',''y'' that could have produced the ''x''-partial derivative 2''x''+''y''.
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| 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.
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| ==See also==
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| <div style="-moz-column-count:2; column-count:2">
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| *[[d'Alembertian operator]]
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| *[[Chain rule]]
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| *[[Curl (mathematics)]]
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| *[[Directional derivative]]
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| *[[Divergence]]
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| *[[Exterior derivative]]
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| *[[Gradient]]
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| *[[Jacobian matrix and determinant]]
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| *[[Laplacian]]
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| *[[Symmetry of second derivatives]]
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| *[[Triple product rule]], also known as the cyclic chain rule.
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| </div>
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| ==Notes==
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| <references />
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| ==External links==
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| *{{springer|title=Partial derivative|id=p/p071620}}
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| *[http://mathworld.wolfram.com/PartialDerivative.html Partial Derivatives] at MathWorld
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| [[Category:Multivariable calculus]]
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| [[Category:Differential operators]]
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