Editing Partial derivative (section)

==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.