Partial derivative

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In mathematics, a partial derivative of a 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 Template:Block indent It can be thought of as the rate of change of the function in the <math>x</math>-direction.

Sometimes, for Template:Nowrap 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">Template:Citation</ref> who used it for 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">Template:Citation</ref>

DefinitionEdit

Like ordinary derivatives, the partial derivative is defined as a limit. Let Template:Mvar be an open subset of <math>\R^n</math> and <math>f:U\to\R</math> a function. The partial derivative of Template:Mvar at the point <math>\mathbf{a}=(a_1, \ldots, a_n) \in U</math> with respect to the Template:Mvar-th variable Template:Math 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 Template:Mvar-th variable Template:Math. Even if all partial derivatives <math>\partial f / \partial x_i(a)</math> exist at a given point Template:Mvar, the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of Template:Mvar and are continuous there, then Template:Mvar is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that Template:Mvar is a Template:Math function. This can be used to generalize for vector valued functions, Template:Nowrap 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 Template:Mvar and can again be partially differentiated. If the direction of derivative is Template:Em repeated, it is called a mixed partial derivative. If all mixed second order partial derivatives are continuous at a point (or on a set), Template:Mvar is termed a Template:Math function at that point (or on that set); in this case, the partial derivatives can be exchanged by 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>

NotationEdit

Template:See For the following examples, let Template:Mvar be a function in Template:Mvar, Template:Mvar, and Template:Mvar.

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 Template:Mvar with respect to Template:Mvar, holding Template:Mvar and Template:Mvar 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 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 Template:Nowrap However, this convention breaks down when we want to evaluate the partial derivative at a point like Template:Nowrap 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 Template:Mvar-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>Template:Cite book</ref>

For higher order partial derivatives, the partial derivative (function) of <math>D_i f</math> with respect to the Template:Mvar-th variable is denoted Template:Nowrap That is, Template:Nowrap 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 implies that <math>D_{i,j}=D_{j,i}</math> as long as comparatively mild regularity conditions on Template:Mvar are satisfied.

GradientEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 Template:Nowrap In this case Template:Mvar has a partial derivative <math>\partial f/\partial x_j</math> with respect to each variable Template:Math. At the point Template:Mvar, 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 Template:Mvar at Template:Mvar. If Template:Mvar is differentiable at every point in some domain, then the gradient is a vector-valued function Template:Math which takes the point Template:Mvar to the vector Template:Math. Consequently, the gradient produces a vector field.

A common abuse of notation is to define the del operator (Template:Math) as follows in three-dimensional Euclidean space <math>\R^3</math> with unit vectors Template:Nowrap, \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 Template:Mvar-dimensional Euclidean space <math>\R^n</math> with coordinates <math>x_1, \ldots, x_n</math> and unit vectors Template:Nowrap_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 derivativeEdit

Template:Excerpt

ExampleEdit

Suppose that Template:Mvar is a function of more than one variable. For instance,

Template:Multiple image

The graph of this function defines a surface in Euclidean space. To every point on this surface, there are an infinite number of tangent lines. 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 Template:Mvar-plane, and those that are parallel to the Template:Mvar-plane (which result from holding either Template:Mvar or Template:Mvar constant, respectively).

To find the slope of the line tangent to the function at Template:Math and parallel to the Template:Mvar-plane, we treat Template:Mvar as a constant. The graph and this plane are shown on the right. Below, we see how the function looks on the plane Template:Math. By finding the derivative of the equation while assuming that Template:Mvar is a constant, we find that the slope of Template:Mvar at the point Template:Math is:

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

So at Template:Math, by substitution, the slope is Template:Math. Therefore,

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

at the point Template:Math. That is, the partial derivative of Template:Mvar with respect to Template:Mvar at Template:Math is Template:Math, as shown in the graph.

The function Template:Mvar can be reinterpreted as a family of functions of one variable indexed by the other variables:

In other words, every value of Template:Mvar defines a function, denoted Template:Math, which is a function of one variable Template:Mvar.<ref>This can also be expressed as the adjointness between the product space and function space constructions.</ref> That is,

In this section the subscript notation Template:Math denotes a function contingent on a fixed value of Template:Mvar, and not a partial derivative.

Once a value of Template:Mvar is chosen, say Template:Mvar, then Template:Math determines a function Template:Math which traces a curve Template:Math on the Template:Mvar-plane:

In this expression, Template:Mvar is a Template:Em, not a Template:Em, so Template:Math is a function of only one real variable, that being Template:Mvar. Consequently, the definition of the derivative for a function of one variable applies:

The above procedure can be performed for any choice of Template:Mvar. Assembling the derivatives together into a function gives a function which describes the variation of Template:Mvar in the Template:Mvar direction:

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

This is the partial derivative of Template:Mvar with respect to Template:Mvar. Here 'Template:Mvar' is a rounded 'd' called the partial derivative symbol; to distinguish it from the letter 'd', 'Template:Mvar' is sometimes pronounced "partial".

Higher order partial derivativesEdit

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 Template:Mvar is simply the partial derivative of the partial derivative (both with respect to Template:Mvar):<ref>Template:Cite book</ref>Template:Rp

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

The cross partial derivative with respect to Template:Mvar and Template:Mvar is obtained by taking the partial derivative of Template:Mvar with respect to Template:Mvar, and then taking the partial derivative of the result with respect to Template:Mvar, 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'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 conditions in optimization problems. The higher order partial derivatives can be obtained by successive differentiation

Antiderivative analogueEdit

There is a concept for partial derivatives that is analogous to antiderivatives 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 Template:Mvar (treating Template:Mvar 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 Template:Mvar. 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 Template:Mvar 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 Template:Nowrap where Template:Mvar is any one-argument function, represents the entire set of functions in variables Template:Math that could have produced the Template:Mvar-partial derivative Template:Nowrap

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.

ApplicationsEdit

GeometryEdit

File:Cone 3d.png

The volume of a cone depends on height and radius

The volume Template:Mvar of a cone depends on the cone's height Template:Mvar and its radius Template:Mvar according to the formula

The partial derivative of Template:Mvar with respect to Template:Mvar 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 Template:Mvar equals Template:Nowrap 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 of Template:Mvar with respect to Template:Mvar and Template:Mvar 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 Template:Mvar,

This gives the total derivative with respect to Template:Mvar,

which simplifies to

Similarly, the total derivative with respect to Template:Mvar is

The total derivative with respect to Template:Em Template:Mvar and Template:Mvar 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>

OptimizationEdit

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 Template:Math with respect to the choice of the quantities Template:Mvar and Template:Mvar of two different types of output. The first order conditions for this optimization are Template:Math. Since both partial derivatives Template:Math and Template:Math will generally themselves be functions of both arguments Template:Mvar and Template:Mvar, these two first order conditions form a system of two equations in two unknowns.

Thermodynamics, quantum mechanics and mathematical physicsEdit

Partial derivatives appear in thermodynamic equations like Gibbs-Duhem equation, in quantum mechanics as in 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 fractions Template:Math in the following example involving the Gibbs energies in a ternary mixture system:

<math display="block">\bar{G_2}= G + (1-x_2) \left(\fracTemplate:\partial G Template:\partial x 2\right)_{\frac{x_1}{x_3}} </math>

Express mole fractions 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 equations 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 resizingEdit

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.

EconomicsEdit

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.

NotesEdit

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External linksEdit

Template:Calculus topics

Partial derivative

Contents

DefinitionEdit

NotationEdit

GradientEdit

Directional derivativeEdit

ExampleEdit

Higher order partial derivativesEdit

Antiderivative analogueEdit

ApplicationsEdit

GeometryEdit

OptimizationEdit

Thermodynamics, quantum mechanics and mathematical physicsEdit

Image resizingEdit

EconomicsEdit

See alsoEdit

NotesEdit

External linksEdit