In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a partial derivative of a
function of several variables is its
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
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
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
.
The partial derivative of a function
with respect to the variable
is variously denoted by
It can be thought of as the rate of change of the function in the
-direction.
Sometimes, for
, the partial derivative of
with respect to
is denoted as
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:
:
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, 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.
Definition
Like ordinary derivatives, the partial derivative is defined as a
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
. Let ''U'' be an
open subset of
and
a function. The partial derivative of ''f'' at the point
with respect to the ''i''-th variable ''x''
''i'' is defined as
:
Even if all partial derivatives ''∂f''/''∂x''
''i''(''a'') exist at a given point ''a'', the function need not be
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
there. However, if all partial derivatives exist in a
neighborhood 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''
1 function. This can be used to generalize for vector valued functions, by carefully using a componentwise argument.
The partial derivative
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
Clairaut's theorem
Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in hydrostatic equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatise ...
:
:
Notation
For the following examples, let
be a function in
and
.
First-order partial derivatives:
:
Second-order partial derivatives:
:
Second-order
mixed derivatives:
:
Higher-order partial and mixed derivatives:
:
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
with respect to
, holding
and
constant, is often expressed as
:
Conventionally, for clarity and simplicity of notation, the partial derivative ''function'' and the ''value'' of the function at a specific point are
conflated
Conflation is the merging of two or more sets of information, texts, ideas, opinions, etc., into one, often in error. Conflation is often misunderstood. It originally meant to fuse or blend, but has since come to mean the same as equate, treati ...
by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like
:
is used for the function, while
:
might be used for the value of the function at the point
. However, this convention breaks down when we want to evaluate the partial derivative at a point like
. In such a case, evaluation of the function must be expressed in an unwieldy manner as
:
or
:
in order to use the Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with
as the partial derivative symbol with respect to the ''i''th variable. For instance, one would write
for the example described above, while the expression
represents the partial derivative ''function'' with respect to the 1st variable.
For higher order partial derivatives, the partial derivative (function) of
with respect to the ''j''th variable is denoted
. That is,
, 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 hydrostatic equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatise ...
implies that
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''
1, ..., ''x
n'') on a domain in Euclidean space
(e.g., on
or
). 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
:
This vector is called the ''
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
'' 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.
A common
abuse of notation is to define the
del operator (∇) as follows in three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
with
unit vectors :
:
Or, more generally, for ''n''-dimensional Euclidean space
with coordinates
and unit vectors
:
:
Directional derivative
Example
Suppose that ''f'' is a function of more than one variable. For instance,
:
.
The
graph of this function defines a
surface in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. 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
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
. Usually, the lines of most interest are those that are parallel to the
-plane, and those that are parallel to the
-plane (which result from holding either
or
constant, respectively).
To find the slope of the line tangent to the function at
and parallel to the
-plane, we treat
as a constant. The graph and this plane are shown on the right. Below, we see how the function looks on the plane
. By finding the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the equation while assuming that
is a constant, we find that the slope of ''
'' at the point
is:
:
So at
, by substitution, the slope is 3. Therefore,
:
at the point
. That is, the partial derivative of
with respect to
at
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:
:
In other words, every value of ''y'' defines a function, denoted ''f
y'', which is a function of one variable ''x''. That is,
:
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
-plane:
:
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:
:
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:
:
This is the partial derivative of ''f'' with respect to ''x''. Here ''∂'' is a rounded ''d'' called the ''
partial derivative symbol
Partial may refer to:
Mathematics
*Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant
** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial ...
''; to distinguish it from the letter ''d'', ''∂'' is sometimes pronounced "partial".
Higher order partial derivatives
Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the function
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.]
:
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
:
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,
:
or equivalently
Own and cross partial derivatives appear in the
Hessian matrix which is used in the
second order condition
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information abo ...
s in
optimization problems.
The higher order partial derivatives can be obtained by successive differentiation
Antiderivative analogue
There is a concept for partial derivatives that is analogous to
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
s for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function.
Consider the example of
:
The "partial" integral can be taken with respect to ''x'' (treating ''y'' as constant, in a similar manner to partial differentiation):
:
Here, the
"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
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
, 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
.
If all the partial derivatives of a function are known (for example, with the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
), 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.
Applications
Geometry
The
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
''V'' of a
cone depends on the cone's
height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is).
For example, "The height of that building is 50 m" or "The height of an airplane in-flight is ab ...
''h'' and its
radius
In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
''r'' according to the formula
:
The partial derivative of ''V'' with respect to ''r'' is
:
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
equals
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
:
and
:
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'',
:
This gives the total derivative with respect to ''r'':
:
which simplifies to:
:
Similarly, the total derivative with respect to ''h'' is:
:
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, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
vector
:
Optimization
Partial derivatives appear in any calculus-based
optimization problem with more than one choice variable. For example, in
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
a firm may wish to maximize
profit π(''x'', ''y'') with respect to the choice of the quantities ''x'' and ''y'' of two different types of output. The
first order condition
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about ...
s for this optimization are π
''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.
Thermodynamics, 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 from
mathematical 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 developm ...
. Here the variables being held constant in partial derivatives can be ratio of simple variables like
mole fractions ''x
i'' in the following example involving the Gibbs energies in a ternary mixture system:
:
Express
mole fractions of a component as functions of other components' mole fraction and binary mole ratios:
:
:
Differential quotients can be formed at constant ratios like those above:
:
:
Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems:
:
:
:
which can be used for solving
partial differential equations like:
:
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
Seam carving (or liquid rescaling) is an algorithm for content-aware image resizing, developed by Shai Avidan, of Mitsubishi Electric Research Laboratories (MERL), and Ariel Shamir, of the Interdisciplinary Center and MERL. It functions by es ...
, these algorithms require each
pixel
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device.
In most digital display devices, pixels are the ...
in an image to be assigned a numerical 'energy' to describe their dissimilarity against orthogonal adjacent pixels. The
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
then progressively removes rows or columns with the lowest energy. The formula established to determine a pixel's energy (magnitude of
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
at a pixel) depends heavily on the constructs of partial derivatives.
Economics
Partial derivatives play a prominent role in
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
, 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.
See also
*
d'Alembertian operator
In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of ...
*
Chain rule
*
Curl (mathematics)
*
Divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
*
Exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
*
Iterated integral
*
Jacobian matrix and determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
*
Laplacian
*
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 real variables, functions of several variables: the Differential calculus, di ...
*
Symmetry of second derivatives
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function
:f\left(x_1,\, x_2,\, \ldots,\, x_n\right)
of ''n ...
*
Triple product rule
The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodyna ...
, also known as the cyclic chain rule.
Notes
References
External links
*
Partial Derivativesat
MathWorld
{{Calculus topics
Multivariable calculus
Differential operators