Editing Partial derivative (section)

== 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".