In

_{''i''} is defined as
:$\backslash begin\; \backslash fracf(\backslash mathbf)\; \&\; =\; \backslash lim\_\; \backslash frac\; \backslash \backslash \; \&\; =\; \backslash lim\_\; \backslash frac\; \backslash end$
Even if all partial derivatives ''∂f''/''∂x''_{''i''}(''a'') exist at a given point ''a'', the function need not be ^{1} function. This can be used to generalize for vector valued functions, by carefully using a componentwise argument.
The partial derivative $\backslash frac$ 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''^{2} function at that point (or on that set); in this case, the partial derivatives can be exchanged by

_{1}, ..., ''x_{n}'') on a domain in Euclidean space $\backslash R^n$ (e.g., on $\backslash R^2$ or $\backslash R^3$). In this case ''f'' has a partial derivative ''∂f''/''∂x_{j}'' with respect to each variable ''x''_{''j''}. At the point ''a'', these partial derivatives define the vector
: $\backslash nabla\; f(a)\; =\; \backslash left(\backslash frac(a),\; \backslash ldots,\; \backslash frac(a)\backslash right).$
This vector is called the ''

_{y}'', which is a function of one variable ''x''. That is,
: $f\_y(x)\; =\; x^2\; +\; xy\; +\; y^2.$
In this section the subscript notation ''f_{y}'' denotes a function contingent on a fixed value of ''y'', and not a partial derivative.
Once a value of ''y'' is chosen, say ''a'', then ''f''(''x'',''y'') determines a function ''f_{a}'' which traces a curve ''x''^{2} + ''ax'' + ''a''^{2} on the $xz$-plane:
: $f\_a(x)\; =\; x^2\; +\; ax\; +\; a^2$.
In this expression, ''a'' is a ''constant'', not a ''variable'', so ''f_{a}'' is a function of only one real variable, that being ''x''. Consequently, the definition of the derivative for a function of one variable applies:
: $f\_a\text{'}(x)\; =\; 2x\; +\; a$.
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 ''x'' direction:
: $\backslash frac(x,y)\; =\; 2x\; +\; y.$
This is the partial derivative of ''f'' with respect to ''x''. Here ''∂'' is a rounded ''d'' called the '' partial derivative symbol''; to distinguish it from the letter ''d'', ''∂'' is sometimes pronounced "partial".

_{''x''} = 0 = π_{''y''}. Since both partial derivatives π_{''x''} and π_{''y''} will generally themselves be functions of both arguments ''x'' and ''y'', these two first order conditions form a system of two equations in two unknowns.

_{i}'' in the following example involving the Gibbs energies in a ternary mixture system:
:$\backslash bar=\; G\; +\; (1-x\_2)\; \backslash left(\backslash frac\backslash right)\_$
Express

gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

at a pixel) depends heavily on the constructs of partial derivatives.

Partial Derivatives

at

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a partial derivative of a function of several variables
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

is its derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

with respect to one of those variables, with the others held constant (as opposed to the total derivative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, in which all variables are allowed to vary). Partial derivatives are used in vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product differentiation, in ...

and differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...

.
The partial derivative of a function $f(x,\; y,\; \backslash dots)$ with respect to the variable $x$ is variously denoted by
: $f\text{'}\_x$, $\backslash partial\_x\; f$, $\backslash \; D\_xf$, $D\_1f$, $\backslash fracf$, or $\backslash frac$.
Sometimes, for $z=f(x,\; y,\; \backslash ldots)$, the partial derivative of $z$ with respect to $x$ is denoted as $\backslash tfrac.$ 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:
:$f\text{'}\_x(x,\; y,\; \backslash ldots),\; \backslash frac\; (x,\; y,\; \backslash ldots).$
The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet
Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (; 17 September 1743 – 29 March 1794), known as Nicolas de Condorcet, was a French philosopher
A philosopher is someone who practices philosophy
Philosophy (from , ) is th ...

from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named a ...

(1786) (although he later abandoned it, Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a Germans, German mathematician who made fundamental contributions to elliptic functions, Dynamics (mechanics), dynamics, differential equations, determinants, and number th ...

reintroduced the symbol in 1841).
Definition

Like ordinary derivatives, the partial derivative is defined as alimit
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. Let ''U'' be an open subset
Open or OPEN may refer to:
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* Open (band)
Open is a band.
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Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...

of $\backslash R^n$ and $f:U\backslash to\backslash R$ a function. The partial derivative of ''f'' at the point $\backslash mathbf=(a\_1,\; \backslash ldots,\; a\_n)\; \backslash in\; U$ with respect to the ''i''-th variable ''x''continuous
Continuity or continuous may refer to:
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* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
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there. However, if all partial derivatives exist in a neighborhood
A neighbourhood (British English
British English (BrE) is the standard dialect
A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...

of ''a'' and are continuous there, then ''f'' is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that ''f'' is a ''C''Clairaut's theorem
Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatise''Théorie ...

:
:$\backslash frac\; =\; \backslash frac\; .$
Notation

For the following examples, let $f$ be a function in $x,\; y$ and $z$. First-order partial derivatives: :$\backslash frac\; =\; f\text{'}\_x\; =\; \backslash partial\_x\; f.$ Second-order partial derivatives: :$\backslash frac\; =\; f\text{'}\text{'}\_\; =\; \backslash partial\_\; f\; =\; \backslash partial\_x^2\; f.$ Second-order mixed derivatives: :$\backslash frac\; =\; \backslash frac\; \backslash left(\; \backslash frac\; \backslash right)\; =\; (f\text{'}\_)\text{'}\_\; =\; f\text{'}\text{'}\_\; =\; \backslash partial\_\; f\; =\; \backslash partial\_y\; \backslash partial\_x\; f\; .$ Higher-order partial and mixed derivatives: :$\backslash frac\; =\; f^\; =\; \backslash partial\_x^i\; \backslash partial\_y^j\; \backslash partial\_z^k\; f.$ 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 asstatistical mechanics
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

, the partial derivative of $f$ with respect to $x$, holding $y$ and $z$ constant, is often expressed as
:$\backslash left(\; \backslash frac\; \backslash right)\_\; .$
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
:$\backslash frac$
is used for the function, while
:$\backslash frac$
might be used for the value of the function at the point $(x,y,z)=(u,v,w)$. However, this convention breaks down when we want to evaluate the partial derivative at a point like $(x,y,z)=(17,\; u+v,\; v^2)$. In such a case, evaluation of the function must be expressed in an unwieldy manner as
:$\backslash frac(17,\; u+v,\; v^2)$
or
:$\backslash left.\; \backslash frac\backslash right\; ,\; \_$
in order to use the Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with $D\_i$ as the partial derivative symbol with respect to the ''i''th variable. For instance, one would write $D\_1\; f(17,\; u+v,\; v^2)$ for the example described above, while the expression $D\_1\; f$ represents the partial derivative ''function'' with respect to the 1st variable.
For higher order partial derivatives, the partial derivative (function) of $D\_i\; f$ with respect to the ''j''th variable is denoted $D\_j(D\_i\; f)=D\_\; f$. That is, $D\_j\backslash circ\; D\_i\; =D\_$, 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
Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatise''Théorie ...

implies that $D\_=D\_$ as long as comparatively mild regularity conditions on ''f'' are satisfied.
Gradient

An important example of a function of several variables is the case of a scalar-valued function ''f''(''x''gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

'' 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
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

.
A common abuse of notation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

is to define the del operator
Del, or nabla, is an operator used in mathematics (particularly in vector calculus
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional E ...

(∇) as follows in three-dimensional Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

$\backslash R^3$ with unit vectors
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

$\backslash hat,\; \backslash hat,\; \backslash hat$:
: $\backslash nabla\; =\; \backslash left;\; href="/html/ALL/s/\backslash right.html"\; ;"title="\backslash right">\backslash right$right
Rights are legal
Law is a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system, surrounded and influenced by its environment, is desc ...

\hat
Or, more generally, for ''n''-dimensional Euclidean space $\backslash R^n$ with coordinates $x\_1,\; \backslash ldots,\; x\_n$ and unit vectors $\backslash hat\_1,\; \backslash ldots,\; \backslash hat\_n$:
: $\backslash nabla\; =\; \backslash sum\_^n\; \backslash left;\; href="/html/ALL/s/frac\_\backslash right.html"\; ;"title="frac\; \backslash right">frac\; \backslash right$
Directional derivative

Example

Suppose that ''f'' is a function of more than one variable. For instance, : $z\; =\; f(x,y)\; =\; x^2\; +\; xy\; +\; y^2$. Thegraph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...

of this function defines a surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile.
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...

in Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

. To every point on this surface, there are an infinite number of tangent line
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

s. Partial differentiation is the act of choosing one of these lines and finding its slope
In mathematics, the slope or gradient of a line
Line, lines, The Line, or LINE may refer to:
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* ''Lines'' (film), a 2016 Greek film
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* ''The Line'' (2009 film)
* ''The Line'', ...

. 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).
To find the slope of the line tangent to the function at $P(1,\; 1)$ and parallel to the $xz$-plane, we treat $y$ as a constant. The graph and this plane are shown on the right. Below, we see how the function looks on the plane $y=1$ . By finding the derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of the equation while assuming that $y$ is a constant, we find that the slope of ''$f$'' at the point $(x,\; y)$ is:
: $\backslash frac\; =\; 2x+y$.
So at $(1,\; 1)$, by substitution, the slope is 3. Therefore,
: $\backslash frac\; =\; 3$
at the point $(1,\; 1)$. That is, the partial derivative of $z$ with respect to $x$ at $(1,\; 1)$ is 3, as shown in the graph.
The function ''f'' can be reinterpreted as a family of functions of one variable indexed by the other variables:
: $f(x,y)\; =\; f\_y(x)\; =\; x^2\; +\; xy\; +\; y^2.$
In other words, every value of ''y'' defines a function, denoted ''fHigher order partial derivatives

Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the function $f(x,\; y,\; ...)$ the "own" second partial derivative with respect to ''x'' is simply the partial derivative of the partial derivative (both with respect to ''x''): Chiang, Alpha C. ''Fundamental Methods of Mathematical Economics'', McGraw-Hill, third edition, 1984. :$\backslash frac\; \backslash equiv\; \backslash partial\; \backslash frac\; \backslash equiv\; \backslash frac\; \backslash equiv\; f\_.$ The cross partial derivative with respect to ''x'' and ''y'' is obtained by taking the partial derivative of ''f'' with respect to ''x'', and then taking the partial derivative of the result with respect to ''y'', to obtain :$\backslash frac\; \backslash equiv\; \backslash partial\; \backslash frac\; \backslash equiv\; \backslash frac\; \backslash equiv\; f\_.$Schwarz's theoremIn mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility under certain conditions (see below) of interchanging the order of taking partial derivatives of a Function (mathematics), funct ...

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,
:$\backslash frac\; =\; \backslash frac$
or equivalently $f\_=f\_.$
Own and cross partial derivatives appear in the Hessian matrix
In mathematic
Mathematics (from Greek: ) includes the study of such topics as quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in te ...

which is used in the second order conditions in optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt=
Mathematical optimization (alter ...

problems.
Antiderivative analogue

There is a concept for partial derivatives that is analogous toantiderivative
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...

s for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function.
Consider the example of
:$\backslash frac\; =\; 2x+y.$
The "partial" integral can be taken with respect to ''x'' (treating ''y'' as constant, in a similar manner to partial differentiation):
:$z\; =\; \backslash int\; \backslash frac\; \backslash ,dx\; =\; x^2\; +\; xy\; +\; g(y)$
Here, the 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 $x$ 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 $x^2\; +\; xy\; +\; g(y)$, 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 $2x\; +\; y$.
If all the partial derivatives of a function are known (for example, with the gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

), 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
Conservatism is an aesthetic
Aesthetics, or esthetics (), is a branch of philosophy that deals with the nature of beauty and taste (sociology), taste, as well as the philosophy of art (its own area of philosophy that comes out of aest ...

.
Applications

Geometry

Thevolume
Volume is a scalar quantity expressing the amount
Quantity or amount is a property that can exist as a multitude
Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

''V'' of a cone
A cone is a three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Greek wikt:πα ...

depends on the cone's height
200px, A cuboid demonstrating the dimensions length, width">length.html" ;"title="cuboid demonstrating the dimensions length">cuboid demonstrating the dimensions length, width, and height.
Height is measure of vertical distance, either vertical ...

''h'' and its radius
In classical geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

''r'' according to the formula
:$V(r,\; h)\; =\; \backslash frac.$
The partial derivative of ''V'' with respect to ''r'' is
:$\backslash frac\; =\; \backslash frac,$
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 $h$ equals $\backslash frac,$ 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 ''V'' with respect to ''r'' and ''h'' are respectively
:$\backslash frac\; =\; \backslash overbrace^\backslash frac\; +\; \backslash overbrace^\backslash frac\backslash frac$
and
:$\backslash frac\; =\; \backslash overbrace^\backslash frac\; +\; \backslash overbrace^\backslash frac\backslash frac$
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 ''k'',
:$k\; =\; \backslash frac\; =\; \backslash frac.$
This gives the total derivative with respect to ''r'':
:$\backslash frac\; =\; \backslash frac\; +\; \backslash frack$
which simplifies to:
:$\backslash frac\; =\; k\; \backslash pi\; r^2$
Similarly, the total derivative with respect to ''h'' is:
:$\backslash frac\; =\; \backslash pi\; r^2$
The total derivative with respect to ''both'' r and h of the volume intended as scalar function of these two variables is given by the gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

vector
:$\backslash nabla\; V\; =\; \backslash left(\backslash frac,\backslash frac\backslash right)\; =\; \backslash left(\backslash frac\backslash pi\; rh,\; \backslash frac\backslash pi\; r^2\backslash right)$.
Optimization

Partial derivatives appear in any calculus-basedoptimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt=
Mathematical optimization (alter ...

problem with more than one choice variable. For example, in economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interact ...

a firm may wish to maximize profit
Profit may refer to:
Business and law
* Profit (accounting)
Profit, in accounting
Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information about economic entity, economic en ...

π(''x'', ''y'') with respect to the choice of the quantities ''x'' and ''y'' of two different types of output. The first order conditionIn calculus, a derivative test uses the derivatives of a function (mathematics), function to locate the stationary point, critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivativ ...

s for this optimization are πThermodynamics, quantum mechanics and mathematical physics

Partial derivatives appear in thermodynamic equations like Gibbs-Duhem equation, in quantum mechanics as Schrodinger wave equation as well in other equations frommathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...

. Here the variables being held constant in partial derivatives can be ratio of simple variables like mole fraction
In chemistry
Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they underg ...

s ''xmole fraction
In chemistry
Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they underg ...

s of a component as functions of other components' mole fraction and binary mole ratios:
:$x\_1\; =\; \backslash frac$
:$x\_3\; =\; \backslash frac$
Differential quotients can be formed at constant ratios like those above:
:$\backslash left(\backslash frac\backslash right)\_\; =\; -\; \backslash frac$
:$\backslash left(\backslash frac\backslash right)\_\; =\; -\; \backslash frac$
Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems:
:$X\; =\; \backslash frac$
:$Y\; =\; \backslash frac$
:$Z\; =\; \backslash frac$
which can be used for solving partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...

s like:
:$\backslash left(\backslash frac\backslash right)\_\; =\; \backslash left(\backslash frac\backslash right)\_$
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 eachpixel
In digital imaging
Digital imaging or digital image acquisition is the creation of a representation of the visual characteristics of an object, such as a physical scene or the interior structure of an object. The term is often assumed to imp ...

in an image to be assigned a numerical 'energy' to describe their dissimilarity against orthogonal adjacent pixels. The algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

then progressively removes rows or columns with the lowest energy. The formula established to determine a pixel's energy (magnitude of Economics

Partial derivatives play a prominent role ineconomics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interact ...

, in which most functions describing economic behaviour posit that the behaviour depends on more than one variable. For example, a societal consumption function
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods a ...

may describe the amount spent on consumer goods as depending on both income and wealth; the marginal propensity to consume
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods an ...

is then the partial derivative of the consumption function with respect to income.
See also

*d'Alembertian operator
In special relativity, electromagnetism and Wave, wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, or box operator is the Laplace operator of Minkowski space. The operator is named after F ...

*Chain rule
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of ...

*Curl (mathematics)
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Produc ...

*Divergence
In vector calculus
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

*Exterior derivative
On a differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-interse ...

*Iterated integral
In multivariable calculus, an iterated integral is the result of applying integrals to a Function (mathematics), function of Function of several real variables, more than one variable (for example f(x,y) or f(x,y,z)) in a way that each of the integr ...

*Jacobian matrix and determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function in several variables is the matrix of all its first-order partial derivative
In mathematics, a partial derivative of a function (mathematics)#MULTIVARIATE FUNCTION, fun ...

*Laplacian
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

*Multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with function of several variables, functions of several variables: the Differential calculus, different ...

*Symmetry of second derivativesIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* Triple product rule, also known as the cyclic chain rule.
Notes

References

External links

*Partial Derivatives

at

MathWorld
''MathWorld'' is an online mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geome ...

{{Calculus topics
Multivariable calculus
Differential operators
{{Cat main, Differential operator
In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider Derivative, differentiation as an abstr ...