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| Boilerpipe Text | 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
with respect to the variable
(analogously for any other variable) is variously denoted by
,
,
,
,
,
, or
.
It is 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,
[
1
]
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.
[
2
]
Like ordinary derivatives, the partial derivative is defined as a
limit
. 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
where
is the
unit vector
of
i
-th variable
x
i
. In fact, the last equality shows that the partial derivative is just the
directional derivative
where the direction is the
-th
standard basis
vector.
Even if all partial derivatives
exist at a given point
a
, the function need not be
continuous
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 component-wise argument.
The partial derivative
is itself a function defined on
U
and can be partially-differentiated again. If the direction of derivative is
not
repeated, it is called a
mixed partial derivative
. 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
:
Further information:
∂
For the following examples, let
f
be a function in
x
,
y
, and
z
.
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
f
with respect to
x
, holding
y
and
z
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
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 first variable.
[
3
]
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
implies that
as long as comparatively mild regularity conditions on
f
are satisfied.
An important example of a function of several variables is the case of a
scalar-valued function
on a domain in Euclidean space
(e.g., on
or
).
In this case
f
has a partial derivative
with respect to each variable
x
j
. At the point
a
, these partial derivatives define the vector
This vector is called the
gradient
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
with
unit vectors
:
Or, more generally, for
n
-dimensional Euclidean space
with coordinates
and unit vectors
:
Directional derivative
[
edit
]
Suppose that
f
is a function of more than one variable. For instance,
A graph of
z
=
x
2
+
xy
+
y
2
. For the partial derivative at
(1, 1)
that leaves
y
constant, the corresponding
tangent
line is parallel to the
xz
-plane.
A slice of the graph above showing the function in the
xz
-plane at
y
= 1
. The two axes are shown here with different scales. The slope of the tangent line is 3.
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
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
of the equation while assuming that
y
is a constant, we find that the slope of
f
at the point
(
x
,
y
)
is:
So at
(1, 1)
, by substitution, the slope is
3
. Therefore,
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:
In other words, every value of
y
defines a function, denoted
f
y
, which is a function of one variable
x
.
[
6
]
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
xz
-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
; to distinguish it from the letter 'd', '
∂
' is sometimes pronounced "partial".
Higher order partial derivatives
[
edit
]
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
):
[
7
]
: 316–318
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 conditions
in
optimization
problems.
The higher order partial derivatives can be obtained by successive differentiation
Antiderivative analogue
[
edit
]
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
The so-called 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
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
,
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
), 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
.
The volume of a cone depends on height and radius
The
volume
V
of a
cone
depends on the cone's
height
h
and its
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
h
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
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
vector
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
π(
x
,
y
)
with respect to the choice of the quantities
x
and
y
of two different types of output. The
first order conditions
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
[
edit
]
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
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.
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.
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.
d'Alembert operator
Chain rule
Curl (mathematics)
Divergence
Exterior derivative
Iterated integral
Jacobian matrix and determinant
Laplace operator
Multivariable calculus
Symmetry of second derivatives
Triple product rule
, also known as the cyclic chain rule.
^
Cajori, Florian (1952),
A History of Mathematical Notations
, vol. 2 (3 ed.), The Open Court Publishing Company, 596
^
Miller, Jeff (n.d.).
"Earliest Uses of Symbols of Calculus"
. In O'Connor, John J.;
Robertson, Edmund F.
(eds.).
MacTutor History of Mathematics archive
.
University of St Andrews
. Retrieved
2023-06-15
.
^
Spivak, M. (1965).
Calculus on Manifolds
. New York: W. A. Benjamin. p. 44.
ISBN
9780805390216
.
^
R. Wrede; M.R. Spiegel (2010).
Advanced Calculus
(3rd ed.). Schaum's Outline Series.
ISBN
978-0-07-162366-7
.
^
The applicability extends to functions over spaces without a
metric
and to
differentiable manifolds
, such as in
general relativity
.
^
This can also be expressed as the
adjointness
between the
product space
and
function space
constructions.
^
Chiang, Alpha C.
(1984).
Fundamental Methods of Mathematical Economics
(3rd ed.). McGraw-Hill.
"Partial derivative"
,
Encyclopedia of Mathematics
,
EMS Press
, 2001 [1994]
Partial Derivatives
at
MathWorld |
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## Contents
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- [(Top)](https://en.wikipedia.org/wiki/Partial_derivative)
- [1 Definition](https://en.wikipedia.org/wiki/Partial_derivative#Definition)
- [2 Notation](https://en.wikipedia.org/wiki/Partial_derivative#Notation)
- [3 Gradient](https://en.wikipedia.org/wiki/Partial_derivative#Gradient)
- [4 Directional derivative](https://en.wikipedia.org/wiki/Partial_derivative#Directional_derivative)
- [5 Example](https://en.wikipedia.org/wiki/Partial_derivative#Example)
- [6 Higher order partial derivatives](https://en.wikipedia.org/wiki/Partial_derivative#Higher_order_partial_derivatives)
- [7 Antiderivative analogue](https://en.wikipedia.org/wiki/Partial_derivative#Antiderivative_analogue)
- [8 Applications](https://en.wikipedia.org/wiki/Partial_derivative#Applications)
Toggle Applications subsection
- [8\.1 Geometry](https://en.wikipedia.org/wiki/Partial_derivative#Geometry)
- [8\.2 Optimization](https://en.wikipedia.org/wiki/Partial_derivative#Optimization)
- [8\.3 Thermodynamics, quantum mechanics and mathematical physics](https://en.wikipedia.org/wiki/Partial_derivative#Thermodynamics,_quantum_mechanics_and_mathematical_physics)
- [8\.4 Image resizing](https://en.wikipedia.org/wiki/Partial_derivative#Image_resizing)
- [8\.5 Economics](https://en.wikipedia.org/wiki/Partial_derivative#Economics)
- [9 See also](https://en.wikipedia.org/wiki/Partial_derivative#See_also)
- [10 Notes](https://en.wikipedia.org/wiki/Partial_derivative#Notes)
- [11 External links](https://en.wikipedia.org/wiki/Partial_derivative#External_links)
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# Partial derivative
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Derivative of a function with multiple variables
| |
|---|
| Part of a series of articles about |
| [Calculus](https://en.wikipedia.org/wiki/Calculus "Calculus") |
| ∫ a b f ′ ( t ) d t \= f ( b ) − f ( a ) {\\displaystyle \\int \_{a}^{b}f'(t)\\,dt=f(b)-f(a)}  |
| |
| [Lists of integrals](https://en.wikipedia.org/wiki/Lists_of_integrals "Lists of integrals") [Integral transform](https://en.wikipedia.org/wiki/Integral_transform "Integral transform") [Leibniz integral rule](https://en.wikipedia.org/wiki/Leibniz_integral_rule "Leibniz integral rule") |
| Definitions |
| [Antiderivative](https://en.wikipedia.org/wiki/Antiderivative "Antiderivative") [Integral](https://en.wikipedia.org/wiki/Integral "Integral") ([improper](https://en.wikipedia.org/wiki/Improper_integral "Improper integral")) [Riemann integral](https://en.wikipedia.org/wiki/Riemann_integral "Riemann integral") [Lebesgue integration](https://en.wikipedia.org/wiki/Lebesgue_integration "Lebesgue integration") [Contour integration](https://en.wikipedia.org/wiki/Contour_integration "Contour integration") [Integral of inverse functions](https://en.wikipedia.org/wiki/Integral_of_inverse_functions "Integral of inverse functions") |
| Integration by |
| [Parts](https://en.wikipedia.org/wiki/Integration_by_parts "Integration by parts") [Discs](https://en.wikipedia.org/wiki/Disc_integration "Disc integration") [Cylindrical shells](https://en.wikipedia.org/wiki/Shell_integration "Shell integration") [Substitution](https://en.wikipedia.org/wiki/Integration_by_substitution "Integration by substitution") ([trigonometric](https://en.wikipedia.org/wiki/Trigonometric_substitution "Trigonometric substitution"), [tangent half-angle](https://en.wikipedia.org/wiki/Tangent_half-angle_substitution "Tangent half-angle substitution"), [Euler](https://en.wikipedia.org/wiki/Euler_substitution "Euler substitution")) [Euler's formula](https://en.wikipedia.org/wiki/Integration_using_Euler%27s_formula "Integration using Euler's formula") [Partial fractions](https://en.wikipedia.org/wiki/Partial_fractions_in_integration "Partial fractions in integration") ([Heaviside's method](https://en.wikipedia.org/wiki/Heaviside_cover-up_method "Heaviside cover-up method")) [Changing order](https://en.wikipedia.org/wiki/Order_of_integration_\(calculus\) "Order of integration (calculus)") [Reduction formulae](https://en.wikipedia.org/wiki/Integration_by_reduction_formulae "Integration by reduction formulae") [Differentiating under the integral sign](https://en.wikipedia.org/wiki/Leibniz_integral_rule#Evaluating_definite_integrals "Leibniz integral rule") [Risch algorithm](https://en.wikipedia.org/wiki/Risch_algorithm "Risch algorithm") |
| |
| [Geometric](https://en.wikipedia.org/wiki/Geometric_series "Geometric series") ([arithmetico-geometric](https://en.wikipedia.org/wiki/Arithmetico%E2%80%93geometric_sequence "Arithmetico–geometric sequence")) [Harmonic](https://en.wikipedia.org/wiki/Harmonic_series_\(mathematics\) "Harmonic series (mathematics)") [Alternating](https://en.wikipedia.org/wiki/Alternating_series "Alternating series") [Power](https://en.wikipedia.org/wiki/Power_series "Power series") [Binomial](https://en.wikipedia.org/wiki/Binomial_series "Binomial series") [Taylor](https://en.wikipedia.org/wiki/Taylor_series "Taylor series") |
| [Convergence tests](https://en.wikipedia.org/wiki/Convergence_tests "Convergence tests") |
| [Summand limit (term test)](https://en.wikipedia.org/wiki/Term_test "Term test") [Ratio](https://en.wikipedia.org/wiki/Ratio_test "Ratio test") [Root](https://en.wikipedia.org/wiki/Root_test "Root test") [Integral](https://en.wikipedia.org/wiki/Integral_test_for_convergence "Integral test for convergence") [Direct comparison](https://en.wikipedia.org/wiki/Direct_comparison_test "Direct comparison test") [Limit comparison](https://en.wikipedia.org/wiki/Limit_comparison_test "Limit comparison test") [Alternating series](https://en.wikipedia.org/wiki/Alternating_series_test "Alternating series test") [Cauchy condensation](https://en.wikipedia.org/wiki/Cauchy_condensation_test "Cauchy condensation test") [Dirichlet](https://en.wikipedia.org/wiki/Dirichlet%27s_test "Dirichlet's test") [Abel](https://en.wikipedia.org/wiki/Abel%27s_test "Abel's test") |
| |
| [Gradient](https://en.wikipedia.org/wiki/Gradient "Gradient") [Divergence](https://en.wikipedia.org/wiki/Divergence "Divergence") [Curl](https://en.wikipedia.org/wiki/Curl_\(mathematics\) "Curl (mathematics)") [Laplacian](https://en.wikipedia.org/wiki/Laplace_operator "Laplace operator") [Directional derivative](https://en.wikipedia.org/wiki/Directional_derivative "Directional derivative") [Identities](https://en.wikipedia.org/wiki/Vector_calculus_identities "Vector calculus identities") |
| Theorems |
| [Gradient](https://en.wikipedia.org/wiki/Gradient_theorem "Gradient theorem") [Green's](https://en.wikipedia.org/wiki/Green%27s_theorem "Green's theorem") [Stokes'](https://en.wikipedia.org/wiki/Stokes%27_theorem "Stokes' theorem") [Divergence](https://en.wikipedia.org/wiki/Divergence_theorem "Divergence theorem") [Generalized Stokes](https://en.wikipedia.org/wiki/Generalized_Stokes_theorem "Generalized Stokes theorem") [Helmholtz decomposition](https://en.wikipedia.org/wiki/Helmholtz_decomposition "Helmholtz decomposition") |
| Formalisms |
| [Matrix](https://en.wikipedia.org/wiki/Matrix_calculus "Matrix calculus") [Tensor](https://en.wikipedia.org/wiki/Tensor_calculus "Tensor calculus") [Exterior](https://en.wikipedia.org/wiki/Exterior_derivative "Exterior derivative") [Geometric](https://en.wikipedia.org/wiki/Geometric_calculus "Geometric calculus") |
| Definitions |
| [Partial derivative]() [Multiple integral](https://en.wikipedia.org/wiki/Multiple_integral "Multiple integral") [Line integral](https://en.wikipedia.org/wiki/Line_integral "Line integral") [Surface integral](https://en.wikipedia.org/wiki/Surface_integral "Surface integral") [Volume integral](https://en.wikipedia.org/wiki/Volume_integral "Volume integral") [Jacobian](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant "Jacobian matrix and determinant") [Hessian](https://en.wikipedia.org/wiki/Hessian_matrix "Hessian matrix") |
| |
| [Calculus on Euclidean space](https://en.wikipedia.org/wiki/Calculus_on_Euclidean_space "Calculus on Euclidean space") [Generalized functions](https://en.wikipedia.org/wiki/Generalized_function "Generalized function") [Limit of distributions](https://en.wikipedia.org/wiki/Limit_of_distributions "Limit of distributions") |
| Specialized [Fractional](https://en.wikipedia.org/wiki/Fractional_calculus "Fractional calculus") [Malliavin](https://en.wikipedia.org/wiki/Malliavin_calculus "Malliavin calculus") [Stochastic](https://en.wikipedia.org/wiki/Stochastic_calculus "Stochastic calculus") [Variations](https://en.wikipedia.org/wiki/Calculus_of_variations "Calculus of variations") |
| Miscellanea [Precalculus](https://en.wikipedia.org/wiki/Precalculus "Precalculus") [History](https://en.wikipedia.org/wiki/History_of_calculus "History of calculus") [Glossary](https://en.wikipedia.org/wiki/Glossary_of_calculus "Glossary of calculus") [List of topics](https://en.wikipedia.org/wiki/List_of_calculus_topics "List of calculus topics") [Integration Bee](https://en.wikipedia.org/wiki/Integration_Bee "Integration Bee") [Mathematical analysis](https://en.wikipedia.org/wiki/Mathematical_analysis "Mathematical analysis") [Nonstandard analysis](https://en.wikipedia.org/wiki/Nonstandard_analysis "Nonstandard analysis") |
| [v](https://en.wikipedia.org/wiki/Template:Calculus "Template:Calculus") [t](https://en.wikipedia.org/wiki/Template_talk:Calculus "Template talk:Calculus") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Calculus "Special:EditPage/Template:Calculus") |
In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), a **partial derivative** of a [function of several variables](https://en.wikipedia.org/wiki/Function_\(mathematics\)#MULTIVARIATE_FUNCTION "Function (mathematics)") is its [derivative](https://en.wikipedia.org/wiki/Derivative "Derivative") with respect to one of those variables, with the others held constant (as opposed to the [total derivative](https://en.wikipedia.org/wiki/Total_derivative "Total derivative"), in which all variables are allowed to vary). Partial derivatives are used in [vector calculus](https://en.wikipedia.org/wiki/Vector_calculus "Vector calculus") and [differential geometry](https://en.wikipedia.org/wiki/Differential_geometry "Differential geometry").
The partial derivative of a function f ( x , y , … ) {\\displaystyle f(x,y,\\dots )}  with respect to the variable x {\\displaystyle x}  (analogously for any other variable) is variously denoted by
f
x
{\\displaystyle f\_{x}}
,
f
x
′
{\\displaystyle f'\_{x}}
,
∂
x
f
{\\displaystyle \\partial \_{x}f}
,
D
x
f
{\\displaystyle \\ D\_{x}f}
,
D
e
1
f
{\\displaystyle D\_{\\mathbf {e} \_{1}}f}
D
1
f
{\\displaystyle D\_{1}f}
,
∂
∂
x
f
{\\displaystyle {\\frac {\\partial }{\\partial x}}f}
, or
∂
f
∂
x
{\\displaystyle {\\frac {\\partial f}{\\partial x}}}
.
It is the rate of change of the function in the x {\\displaystyle x} \-direction.
Sometimes, for z \= f ( x , y , … ) {\\displaystyle z=f(x,y,\\ldots )}  , the partial derivative of z {\\displaystyle z}  with respect to x {\\displaystyle x}  is denoted as ∂ z ∂ x . {\\displaystyle {\\tfrac {\\partial z}{\\partial x}}.}  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 x ′ ( x , y , … ) , ∂ f ∂ x ( x , y , … ) . {\\displaystyle f'\_{x}(x,y,\\ldots ),{\\frac {\\partial f}{\\partial x}}(x,y,\\ldots ).} 
The symbol used to denote partial derivatives is [∂](https://en.wikipedia.org/wiki/%E2%88%82 "∂"). One of the first known uses of this symbol in mathematics is by [Marquis de Condorcet](https://en.wikipedia.org/wiki/Marquis_de_Condorcet "Marquis de Condorcet") from 1770,[\[1\]](https://en.wikipedia.org/wiki/Partial_derivative#cite_note-Cajori_History_V2-1) who used it for [partial differences](https://en.wikipedia.org/wiki/Partial_difference_equation "Partial difference equation"). The modern partial derivative notation was created by [Adrien-Marie Legendre](https://en.wikipedia.org/wiki/Adrien-Marie_Legendre "Adrien-Marie Legendre") (1786), although he later abandoned it; [Carl Gustav Jacob Jacobi](https://en.wikipedia.org/wiki/Carl_Gustav_Jacob_Jacobi "Carl Gustav Jacob Jacobi") reintroduced the symbol in 1841.[\[2\]](https://en.wikipedia.org/wiki/Partial_derivative#cite_note-miller_earliest-2)
## Definition
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=1 "Edit section: Definition")\]
Like ordinary derivatives, the partial derivative is defined as a [limit](https://en.wikipedia.org/wiki/Limit_of_a_function "Limit of a function"). Let U be an [open subset](https://en.wikipedia.org/wiki/Open_set "Open set") of R n {\\displaystyle \\mathbb {R} ^{n}}  and f : U → R {\\displaystyle f:U\\to \\mathbb {R} }  a function. The partial derivative of f at the point a \= ( a 1 , … , a n ) ∈ U {\\displaystyle \\mathbf {a} =(a\_{1},\\ldots ,a\_{n})\\in U}  with respect to the i\-th variable *x**i* is defined as
∂ ∂ x i f ( a ) \= lim h → 0 f ( a 1 , … , a i − 1 , a i \+ h , a i \+ 1 … , a n ) − f ( a 1 , … , a i , … , a n ) h \= lim h → 0 f ( a \+ h e i ) − f ( a ) h {\\displaystyle {\\begin{aligned}{\\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{aligned}}} 
where e i {\\displaystyle \\mathbf {e\_{i}} }  is the [unit vector](https://en.wikipedia.org/wiki/Unit_vector "Unit vector") of i\-th variable *x**i*. In fact, the last equality shows that the partial derivative is just the [directional derivative](https://en.wikipedia.org/wiki/Directional_derivative "Directional derivative") where the direction is the i {\\displaystyle i} \-th [standard basis](https://en.wikipedia.org/wiki/Standard_basis "Standard basis") vector.
Even if all partial derivatives ∂ f / ∂ x i ( a ) {\\displaystyle \\partial f/\\partial x\_{i}(a)}  exist at a given point a, the function need not be [continuous](https://en.wikipedia.org/wiki/Continuous_function "Continuous function") there. However, if all partial derivatives exist in a [neighborhood](https://en.wikipedia.org/wiki/Neighborhood_\(topology\) "Neighborhood (topology)") of a and are continuous there, then f is [totally differentiable](https://en.wikipedia.org/wiki/Total_derivative "Total derivative") 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, f : U → R m {\\displaystyle f:U\\to \\mathbb {R} ^{m}}  , by carefully using a component-wise argument.
The partial derivative ∂ f ∂ x {\\textstyle {\\frac {\\partial f}{\\partial x}}}  is itself a function defined on U and can be partially-differentiated again. If the direction of derivative is *not* repeated, it is called a ***mixed partial derivative***. 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](https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Schwarz's_theorem "Symmetry of second derivatives"):
∂ 2 f ∂ x i ∂ x j \= ∂ 2 f ∂ x j ∂ x i . {\\displaystyle {\\frac {\\partial ^{2}f}{\\partial x\_{i}\\partial x\_{j}}}={\\frac {\\partial ^{2}f}{\\partial x\_{j}\\partial x\_{i}}}.} 
## Notation
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=2 "Edit section: Notation")\]
Further information: [∂](https://en.wikipedia.org/wiki/%E2%88%82 "∂")
For the following examples, let f be a function in x, y, and z.
First-order partial derivatives:
∂ f ∂ x \= f x ′ \= ∂ x f . {\\displaystyle {\\frac {\\partial f}{\\partial x}}=f'\_{x}=\\partial \_{x}f.} 
Second-order partial derivatives:
∂ 2 f ∂ x 2 \= f x x ″ \= ∂ x x f \= ∂ x 2 f . {\\displaystyle {\\frac {\\partial ^{2}f}{\\partial x^{2}}}=f''\_{xx}=\\partial \_{xx}f=\\partial \_{x}^{2}f.} 
Second-order [mixed derivatives](https://en.wikipedia.org/wiki/Mixed_derivatives "Mixed derivatives"):
∂ 2 f ∂ y ∂ x \= ∂ ∂ y ( ∂ f ∂ x ) \= ( f x ′ ) y ′ \= f x y ″ \= ∂ y x f \= ∂ y ∂ x f . {\\displaystyle {\\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.} 
Higher-order partial and mixed derivatives:
∂ i \+ j \+ k f ∂ x i ∂ y j ∂ z k \= f ( i , j , k ) \= ∂ x i ∂ y j ∂ z k f . {\\displaystyle {\\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.} 
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](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics"), the partial derivative of f with respect to x, holding y and z constant, is often expressed as
( ∂ f ∂ x ) y , z . {\\displaystyle \\left({\\frac {\\partial f}{\\partial x}}\\right)\_{y,z}.} 
Conventionally, for clarity and simplicity of notation, the partial derivative *function* and the *value* of the function at a specific point are [conflated](https://en.wikipedia.org/wiki/Abuse_of_notation "Abuse of notation") by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like
∂ f ( x , y , z ) ∂ x {\\displaystyle {\\frac {\\partial f(x,y,z)}{\\partial x}}} 
is used for the function, while
∂ f ( u , v , w ) ∂ u {\\displaystyle {\\frac {\\partial f(u,v,w)}{\\partial u}}} 
might be used for the value of the function at the point ( x , y , z ) \= ( u , v , w ) {\\displaystyle (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 ) {\\displaystyle (x,y,z)=(17,u+v,v^{2})}  . In such a case, evaluation of the function must be expressed in an unwieldy manner as
∂ f ( x , y , z ) ∂ x ( 17 , u \+ v , v 2 ) {\\displaystyle {\\frac {\\partial f(x,y,z)}{\\partial x}}(17,u+v,v^{2})} 
or
∂ f ( x , y , z ) ∂ x \| ( x , y , z ) \= ( 17 , u \+ v , v 2 ) {\\displaystyle \\left.{\\frac {\\partial f(x,y,z)}{\\partial x}}\\right\|\_{(x,y,z)=(17,u+v,v^{2})}} 
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 {\\displaystyle 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 ) {\\displaystyle D\_{1}f(17,u+v,v^{2})}  for the example described above, while the expression D 1 f {\\displaystyle D\_{1}f}  represents the partial derivative *function* with respect to the first variable.[\[3\]](https://en.wikipedia.org/wiki/Partial_derivative#cite_note-3)
For higher order partial derivatives, the partial derivative (function) of D i f {\\displaystyle D\_{i}f}  with respect to the j\-th variable is denoted D j ( D i f ) \= D i , j f {\\displaystyle D\_{j}(D\_{i}f)=D\_{i,j}f}  . That is, D j ∘ D i \= D i , j {\\displaystyle D\_{j}\\circ D\_{i}=D\_{i,j}}  , 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](https://en.wikipedia.org/wiki/Clairaut%27s_theorem_on_equality_of_mixed_partials "Clairaut's theorem on equality of mixed partials") implies that D i , j \= D j , i {\\displaystyle D\_{i,j}=D\_{j,i}}  as long as comparatively mild regularity conditions on f are satisfied.
## Gradient
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=3 "Edit section: Gradient")\]
Main article: [Gradient](https://en.wikipedia.org/wiki/Gradient "Gradient")
An important example of a function of several variables is the case of a [scalar-valued function](https://en.wikipedia.org/wiki/Scalar-valued_function "Scalar-valued function") f ( x 1 , … , x n ) {\\displaystyle f(x\_{1},\\ldots ,x\_{n})}  on a domain in Euclidean space R n {\\displaystyle \\mathbb {R} ^{n}}  (e.g., on R 2 {\\displaystyle \\mathbb {R} ^{2}}  or R 3 {\\displaystyle \\mathbb {R} ^{3}}  ). In this case f has a partial derivative ∂ f / ∂ x j {\\displaystyle \\partial f/\\partial x\_{j}}  with respect to each variable *x**j*. At the point a, these partial derivatives define the vector
∇ f ( a ) \= ( ∂ f ∂ x 1 ( a ) , … , ∂ f ∂ x n ( a ) ) . {\\displaystyle \\nabla f(a)=\\left({\\frac {\\partial f}{\\partial x\_{1}}}(a),\\ldots ,{\\frac {\\partial f}{\\partial x\_{n}}}(a)\\right).} 
This vector is called the *[gradient](https://en.wikipedia.org/wiki/Gradient "Gradient")* 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](https://en.wikipedia.org/wiki/Vector_field "Vector field").
A common [abuse of notation](https://en.wikipedia.org/wiki/Abuse_of_notation "Abuse of notation") is to define the [del operator](https://en.wikipedia.org/wiki/Del_operator "Del operator") (∇) as follows in three-dimensional [Euclidean space](https://en.wikipedia.org/wiki/Euclidean_space "Euclidean space") R 3 {\\displaystyle \\mathbb {R} ^{3}}  with [unit vectors](https://en.wikipedia.org/wiki/Unit_vectors "Unit vectors") i ^ , j ^ , k ^ {\\displaystyle {\\hat {\\mathbf {i} }},{\\hat {\\mathbf {j} }},{\\hat {\\mathbf {k} }}}  :
∇ \= \[ ∂ ∂ x \] i ^ \+ \[ ∂ ∂ y \] j ^ \+ \[ ∂ ∂ z \] k ^ {\\displaystyle \\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} }}} ![{\\displaystyle \\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} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c70b5bce4676294a2be68361d333b3f44ce478)
Or, more generally, for n\-dimensional Euclidean space R n {\\displaystyle \\mathbb {R} ^{n}}  with coordinates x 1 , … , x n {\\displaystyle x\_{1},\\ldots ,x\_{n}}  and unit vectors e ^ 1 , … , e ^ n {\\displaystyle {\\hat {\\mathbf {e} }}\_{1},\\ldots ,{\\hat {\\mathbf {e} }}\_{n}}  :
∇ \= ∑ j \= 1 n \[ ∂ ∂ x j \] e ^ j \= \[ ∂ ∂ x 1 \] e ^ 1 \+ \[ ∂ ∂ x 2 \] e ^ 2 \+ ⋯ \+ \[ ∂ ∂ x n \] e ^ n {\\displaystyle \\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}} ![{\\displaystyle \\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4449e767ec489248b3a285415bf722bd550932c7)
## Directional derivative
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This section is an excerpt from [Directional derivative § Definition](https://en.wikipedia.org/wiki/Directional_derivative#Definition "Directional derivative").\[[edit](https://en.wikipedia.org/w/index.php?title=Directional_derivative&action=edit)\]
[](https://en.wikipedia.org/wiki/File:Directional_derivative_contour_plot.svg)
A [contour plot](https://en.wikipedia.org/wiki/Contour_plot "Contour plot") of
f
(
x
,
y
)
\=
x
2
\+
y
2
{\\displaystyle f(x,y)=x^{2}+y^{2}}
, showing the gradient vector in black, and the unit vector
u
{\\displaystyle \\mathbf {u} }
scaled by the directional derivative in the direction of
u
{\\displaystyle \\mathbf {u} }
in orange. The gradient vector is longer because the gradient points in the direction of greatest rate of increase of a function.
The *directional derivative* of a [scalar function](https://en.wikipedia.org/wiki/Scalar_function "Scalar function") f ( x ) \= f ( x 1 , x 2 , … , x n ) {\\displaystyle f(\\mathbf {x} )=f(x\_{1},x\_{2},\\ldots ,x\_{n})}  along a vector v \= ( v 1 , … , v n ) {\\displaystyle \\mathbf {v} =(v\_{1},\\ldots ,v\_{n})}  is the [function](https://en.wikipedia.org/wiki/Function_\(mathematics\) "Function (mathematics)") ∇ v f {\\displaystyle \\nabla \_{\\mathbf {v} }{f}}  defined by the [limit](https://en.wikipedia.org/wiki/Limit_\(mathematics\) "Limit (mathematics)")[\[4\]](https://en.wikipedia.org/wiki/Partial_derivative#cite_note-4) ∇ v f ( x ) \= lim h → 0 f ( x \+ h v ) − f ( x ) h \| \| v \| \| \= 1 \| \| v \| \| d d t f ( x \+ t v ) \| t \= 0 . {\\displaystyle \\nabla \_{\\mathbf {v} }{f}(\\mathbf {x} )=\\lim \_{h\\to 0}{\\frac {f(\\mathbf {x} +h\\mathbf {v} )-f(\\mathbf {x} )}{h\|\|\\mathbf {v} \|\|}}=\\left.{\\frac {1}{\|\|\\mathbf {v} \|\|}}{\\frac {\\mathrm {d} }{\\mathrm {d} t}}f(\\mathbf {x} +t\\mathbf {v} )\\right\|\_{t=0}.} 
This definition is valid in a broad range of contexts, for example, where the [norm](https://en.wikipedia.org/wiki/Euclidean_norm "Euclidean norm") of a vector (and hence a unit vector) is defined.[\[5\]](https://en.wikipedia.org/wiki/Partial_derivative#cite_note-5)
## Example
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Suppose that f is a function of more than one variable. For instance,
z \= f ( x , y ) \= x 2 \+ x y \+ y 2 . {\\displaystyle z=f(x,y)=x^{2}+xy+y^{2}.} 
[](https://en.wikipedia.org/wiki/File:Partial_func_eg.svg)
A graph of *z* = *x*2 + *xy* + *y*2. For the partial derivative at (1, 1) that leaves y constant, the corresponding [tangent](https://en.wikipedia.org/wiki/Tangent "Tangent") line is parallel to the xz\-plane.
[](https://en.wikipedia.org/wiki/File:X2%2BX%2B1.svg)
A slice of the graph above showing the function in the xz\-plane at *y* = 1. The two axes are shown here with different scales. The slope of the tangent line is 3.
The [graph](https://en.wikipedia.org/wiki/Graph_of_a_function "Graph of a function") of this function defines a [surface](https://en.wikipedia.org/wiki/Surface_\(topology\) "Surface (topology)") in [Euclidean space](https://en.wikipedia.org/wiki/Euclidean_space "Euclidean space"). To every point on this surface, there are an infinite number of [tangent lines](https://en.wikipedia.org/wiki/Tangent_line "Tangent line"). Partial differentiation is the act of choosing one of these lines and finding its [slope](https://en.wikipedia.org/wiki/Slope "Slope"). 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](https://en.wikipedia.org/wiki/Derivative "Derivative") of the equation while assuming that y is a constant, we find that the slope of f at the point (*x*, *y*) is:
∂ z ∂ x \= 2 x \+ y . {\\displaystyle {\\frac {\\partial z}{\\partial x}}=2x+y.} 
So at (1, 1), by substitution, the slope is 3. Therefore,
∂ z ∂ x \= 3 {\\displaystyle {\\frac {\\partial z}{\\partial x}}=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 \+ x y \+ y 2 . {\\displaystyle f(x,y)=f\_{y}(x)=x^{2}+xy+y^{2}.} 
In other words, every value of y defines a function, denoted *fy*, which is a function of one variable x.[\[6\]](https://en.wikipedia.org/wiki/Partial_derivative#cite_note-6) That is,
f y ( x ) \= x 2 \+ x y \+ y 2 . {\\displaystyle f\_{y}(x)=x^{2}+xy+y^{2}.} 
In this section the subscript notation *fy* 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 *fa* which traces a curve *x*2 + *ax* + *a*2 on the xz\-plane:
f a ( x ) \= x 2 \+ a x \+ a 2 . {\\displaystyle f\_{a}(x)=x^{2}+ax+a^{2}.} 
In this expression, a is a *constant*, not a *variable*, so *fa* 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 ′ ( x ) \= 2 x \+ a . {\\displaystyle f\_{a}'(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:
∂ f ∂ x ( x , y ) \= 2 x \+ y . {\\displaystyle {\\frac {\\partial f}{\\partial x}}(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](https://en.wikipedia.org/wiki/Partial_derivative_symbol "Partial derivative symbol")*; to distinguish it from the letter 'd', '∂' is sometimes pronounced "partial".
## Higher order partial derivatives
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Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the function f ( x , y , . . . ) {\\displaystyle 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):[\[7\]](https://en.wikipedia.org/wiki/Partial_derivative#cite_note-7): 316–318
∂ 2 f ∂ x 2 ≡ ∂ ∂ f / ∂ x ∂ x ≡ ∂ f x ∂ x ≡ f x x . {\\displaystyle {\\frac {\\partial ^{2}f}{\\partial x^{2}}}\\equiv \\partial {\\frac {\\partial f/\\partial x}{\\partial x}}\\equiv {\\frac {\\partial f\_{x}}{\\partial x}}\\equiv f\_{xx}.} 
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
∂ 2 f ∂ y ∂ x ≡ ∂ ∂ f / ∂ x ∂ y ≡ ∂ f x ∂ y ≡ f x y . {\\displaystyle {\\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}.} 
[Schwarz's theorem](https://en.wikipedia.org/wiki/Schwarz_theorem "Schwarz 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,
∂ 2 f ∂ x ∂ y \= ∂ 2 f ∂ y ∂ x {\\displaystyle {\\frac {\\partial ^{2}f}{\\partial x\\,\\partial y}}={\\frac {\\partial ^{2}f}{\\partial y\\,\\partial x}}} 
or equivalently f y x \= f x y . {\\displaystyle f\_{yx}=f\_{xy}.} 
Own and cross partial derivatives appear in the [Hessian matrix](https://en.wikipedia.org/wiki/Hessian_matrix "Hessian matrix") which is used in the [second order conditions](https://en.wikipedia.org/wiki/Second_order_condition "Second order condition") in [optimization](https://en.wikipedia.org/wiki/Optimization "Optimization") problems. The higher order partial derivatives can be obtained by successive differentiation
## Antiderivative analogue
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There is a concept for partial derivatives that is analogous to [antiderivatives](https://en.wikipedia.org/wiki/Antiderivative "Antiderivative") for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function.
Consider the example of
∂ z ∂ x \= 2 x \+ y . {\\displaystyle {\\frac {\\partial z}{\\partial x}}=2x+y.} 
The so-called partial integral can be taken with respect to x (treating y as constant, in a similar manner to partial differentiation):
z \= ∫ ∂ z ∂ x d x \= x 2 \+ x y \+ g ( y ) . {\\displaystyle z=\\int {\\frac {\\partial z}{\\partial x}}\\,dx=x^{2}+xy+g(y).} 
Here, the [constant of integration](https://en.wikipedia.org/wiki/Constant_of_integration "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 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 \+ x y \+ g ( y ) {\\displaystyle 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 2 x \+ y {\\displaystyle 2x+y}  .
If all the partial derivatives of a function are known (for example, with the [gradient](https://en.wikipedia.org/wiki/Gradient "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](https://en.wikipedia.org/wiki/Conservative_vector_field "Conservative vector field").
## Applications
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=8 "Edit section: Applications")\]
### Geometry
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=9 "Edit section: Geometry")\]
[](https://en.wikipedia.org/wiki/File:Cone_3d.png)
The volume of a cone depends on height and radius
The [volume](https://en.wikipedia.org/wiki/Volume "Volume") V of a [cone](https://en.wikipedia.org/wiki/Cone_\(geometry\) "Cone (geometry)") depends on the cone's [height](https://en.wikipedia.org/wiki/Height "Height") h and its [radius](https://en.wikipedia.org/wiki/Radius "Radius") r according to the formula
V ( r , h ) \= π r 2 h 3 . {\\displaystyle V(r,h)={\\frac {\\pi r^{2}h}{3}}.} 
The partial derivative of V with respect to r is
∂ V ∂ r \= 2 π r h 3 , {\\displaystyle {\\frac {\\partial V}{\\partial r}}={\\frac {2\\pi rh}{3}},} 
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 1 3 π r 2 {\\textstyle {\\frac {1}{3}}\\pi r^{2}}  , 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](https://en.wikipedia.org/wiki/Total_derivative "Total derivative") of V with respect to r and h are respectively
d V d r \= 2 π r h 3 ⏞ ∂ V ∂ r \+ π r 2 3 ⏞ ∂ V ∂ h d h d r , d V d h \= π r 2 3 ⏞ ∂ V ∂ h \+ 2 π r h 3 ⏞ ∂ V ∂ r d r d h . {\\displaystyle {\\begin{aligned}{\\frac {dV}{dr}}&=\\overbrace {\\frac {2\\pi rh}{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 rh}{3}} ^{\\frac {\\partial V}{\\partial r}}{\\frac {dr}{dh}}\\,.\\end{aligned}}} 
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 \= h r \= d h d r . {\\displaystyle k={\\frac {h}{r}}={\\frac {dh}{dr}}.} 
This gives the total derivative with respect to r,
d V d r \= 2 π r h 3 \+ π r 2 3 k , {\\displaystyle {\\frac {dV}{dr}}={\\frac {2\\pi rh}{3}}+{\\frac {\\pi r^{2}}{3}}k\\,,} 
which simplifies to
d V d r \= k π r 2 , {\\displaystyle {\\frac {dV}{dr}}=k\\pi r^{2},} 
Similarly, the total derivative with respect to h is
d V d h \= π r 2 . {\\displaystyle {\\frac {dV}{dh}}=\\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](https://en.wikipedia.org/wiki/Gradient "Gradient") vector
∇ V \= ( ∂ V ∂ r , ∂ V ∂ h ) \= ( 2 3 π r h , 1 3 π r 2 ) . {\\displaystyle \\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).} 
### Optimization
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=10 "Edit section: Optimization")\]
Partial derivatives appear in any calculus-based [optimization](https://en.wikipedia.org/wiki/Optimization "Optimization") problem with more than one choice variable. For example, in [economics](https://en.wikipedia.org/wiki/Economics "Economics") a firm may wish to maximize [profit](https://en.wikipedia.org/wiki/Profit_\(economics\) "Profit (economics)") π(*x*, *y*) with respect to the choice of the quantities x and y of two different types of output. The [first order conditions](https://en.wikipedia.org/wiki/First_order_condition "First order condition") 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](https://en.wikipedia.org/wiki/System_of_equations "System of equations").
### Thermodynamics, quantum mechanics and mathematical physics
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Partial derivatives appear in thermodynamic equations like [Gibbs-Duhem equation](https://en.wikipedia.org/wiki/Gibbs-Duhem_equation "Gibbs-Duhem equation"), in quantum mechanics as in [Schrödinger wave equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation"), as well as in other equations from [mathematical physics](https://en.wikipedia.org/wiki/Mathematical_physics "Mathematical physics"). The variables being held constant in partial derivatives here can be ratios of simple variables like [mole fractions](https://en.wikipedia.org/wiki/Mole_fraction "Mole fraction") *xi* in the following example involving the Gibbs energies in a ternary mixture system:
G 2 ¯ \= G \+ ( 1 − x 2 ) ( ∂ G ∂ x 2 ) x 1 x 3 {\\displaystyle {\\bar {G\_{2}}}=G+(1-x\_{2})\\left({\\frac {\\partial G}{\\partial x\_{2}}}\\right)\_{\\frac {x\_{1}}{x\_{3}}}} 
Express [mole fractions](https://en.wikipedia.org/wiki/Mole_fraction "Mole fraction") of a component as functions of other components' mole fraction and binary mole ratios:
x 1 \= 1 − x 2 1 \+ x 3 x 1 x 3 \= 1 − x 2 1 \+ x 1 x 3 {\\textstyle {\\begin{aligned}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{aligned}}} 
Differential quotients can be formed at constant ratios like those above:
( ∂ x 1 ∂ x 2 ) x 1 x 3 \= − x 1 1 − x 2 ( ∂ x 3 ∂ x 2 ) x 1 x 3 \= − x 3 1 − x 2 {\\displaystyle {\\begin{aligned}\\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{aligned}}} 
Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems:
X \= x 3 x 1 \+ x 3 Y \= x 3 x 2 \+ x 3 Z \= x 2 x 1 \+ x 2 {\\displaystyle {\\begin{aligned}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{aligned}}} 
which can be used for solving [partial differential equations](https://en.wikipedia.org/wiki/Partial_differential_equation "Partial differential equation") like:
( ∂ μ 2 ∂ n 1 ) n 2 , n 3 \= ( ∂ μ 1 ∂ n 2 ) n 1 , n 3 {\\displaystyle \\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}}} 
This equality can be rearranged to have differential quotient of mole fractions on one side.
### Image resizing
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=12 "Edit section: Image resizing")\]
Partial derivatives are key to target-aware image resizing algorithms. Widely known as [seam carving](https://en.wikipedia.org/wiki/Seam_carving "Seam carving"), these algorithms require each [pixel](https://en.wikipedia.org/wiki/Pixel "Pixel") in an image to be assigned a numerical 'energy' to describe their dissimilarity against orthogonal adjacent pixels. The [algorithm](https://en.wikipedia.org/wiki/Algorithm "Algorithm") then progressively removes rows or columns with the lowest energy. The formula established to determine a pixel's energy (magnitude of [gradient](https://en.wikipedia.org/wiki/Gradient "Gradient") at a pixel) depends heavily on the constructs of partial derivatives.
### Economics
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=13 "Edit section: Economics")\]
Partial derivatives play a prominent role in [economics](https://en.wikipedia.org/wiki/Economics "Economics"), in which most functions describing economic behaviour posit that the behaviour depends on more than one variable. For example, a societal [consumption function](https://en.wikipedia.org/wiki/Consumption_function "Consumption function") may describe the amount spent on consumer goods as depending on both income and wealth; the [marginal propensity to consume](https://en.wikipedia.org/wiki/Marginal_propensity_to_consume "Marginal propensity to consume") is then the partial derivative of the consumption function with respect to income.
## See also
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- [d'Alembert operator](https://en.wikipedia.org/wiki/D%27Alembert_operator "D'Alembert operator")
- [Chain rule](https://en.wikipedia.org/wiki/Chain_rule "Chain rule")
- [Curl (mathematics)](https://en.wikipedia.org/wiki/Curl_\(mathematics\) "Curl (mathematics)")
- [Divergence](https://en.wikipedia.org/wiki/Divergence "Divergence")
- [Exterior derivative](https://en.wikipedia.org/wiki/Exterior_derivative "Exterior derivative")
- [Iterated integral](https://en.wikipedia.org/wiki/Iterated_integral "Iterated integral")
- [Jacobian matrix and determinant](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant "Jacobian matrix and determinant")
- [Laplace operator](https://en.wikipedia.org/wiki/Laplace_operator "Laplace operator")
- [Multivariable calculus](https://en.wikipedia.org/wiki/Multivariable_calculus "Multivariable calculus")
- [Symmetry of second derivatives](https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives "Symmetry of second derivatives")
- [Triple product rule](https://en.wikipedia.org/wiki/Triple_product_rule "Triple product rule"), also known as the cyclic chain rule.
## Notes
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1. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-Cajori_History_V2_1-0)**
Cajori, Florian (1952), [*A History of Mathematical Notations*](https://archive.org/details/AHistoryOfMathematicalNotationVolII/page/n153/mode/2up), vol. 2 (3 ed.), The Open Court Publishing Company, 596
2. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-miller_earliest_2-0)**
Miller, Jeff (n.d.). ["Earliest Uses of Symbols of Calculus"](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/calculus/). In O'Connor, John J.; [Robertson, Edmund F.](https://en.wikipedia.org/wiki/Edmund_F._Robertson "Edmund F. Robertson") (eds.). *[MacTutor History of Mathematics archive](https://en.wikipedia.org/wiki/MacTutor_History_of_Mathematics_archive "MacTutor History of Mathematics archive")*. [University of St Andrews](https://en.wikipedia.org/wiki/University_of_St_Andrews "University of St Andrews"). Retrieved 2023-06-15.
3. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-3)**
Spivak, M. (1965). [*Calculus on Manifolds*](https://archive.org/details/SpivakM.CalculusOnManifoldsPerseus2006Reprint). New York: W. A. Benjamin. p. 44. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9780805390216](https://en.wikipedia.org/wiki/Special:BookSources/9780805390216 "Special:BookSources/9780805390216")
.
4. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-4)**
R. Wrede; M.R. Spiegel (2010). *Advanced Calculus* (3rd ed.). Schaum's Outline Series. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-162366-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-162366-7 "Special:BookSources/978-0-07-162366-7")
.
5. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-5)** The applicability extends to functions over spaces without a [metric](https://en.wikipedia.org/wiki/Metric_\(mathematics\) "Metric (mathematics)") and to [differentiable manifolds](https://en.wikipedia.org/wiki/Differentiable_manifold "Differentiable manifold"), such as in [general relativity](https://en.wikipedia.org/wiki/General_relativity "General relativity").
6. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-6)** This can also be expressed as the [adjointness](https://en.wikipedia.org/wiki/Adjoint_functors "Adjoint functors") between the [product space](https://en.wikipedia.org/wiki/Product_topology "Product topology") and [function space](https://en.wikipedia.org/wiki/Function_space "Function space") constructions.
7. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-7)**
[Chiang, Alpha C.](https://en.wikipedia.org/wiki/Alpha_Chiang "Alpha Chiang") (1984). *Fundamental Methods of Mathematical Economics* (3rd ed.). McGraw-Hill.
## External links
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=16 "Edit section: External links")\]
- ["Partial derivative"](https://www.encyclopediaofmath.org/index.php?title=Partial_derivative), *[Encyclopedia of Mathematics](https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics "Encyclopedia of Mathematics")*, [EMS Press](https://en.wikipedia.org/wiki/European_Mathematical_Society "European Mathematical Society"), 2001 \[1994\]
- [Partial Derivatives](http://mathworld.wolfram.com/PartialDerivative.html) at [MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")
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| [Differential calculus](https://en.wikipedia.org/wiki/Differential_calculus "Differential calculus") | [Derivative](https://en.wikipedia.org/wiki/Derivative "Derivative") [Second derivative](https://en.wikipedia.org/wiki/Second_derivative "Second derivative") [Partial derivative]() [Differential](https://en.wikipedia.org/wiki/Differential_\(mathematics\) "Differential (mathematics)") [Differential operator](https://en.wikipedia.org/wiki/Differential_operator "Differential operator") [Mean value theorem](https://en.wikipedia.org/wiki/Mean_value_theorem "Mean value theorem") [Notation](https://en.wikipedia.org/wiki/Notation_for_differentiation "Notation for differentiation") [Leibniz's notation](https://en.wikipedia.org/wiki/Leibniz%27s_notation "Leibniz's notation") [Newton's notation](https://en.wikipedia.org/wiki/Newton%27s_notation_for_differentiation "Newton's notation for differentiation") [Rules of differentiation](https://en.wikipedia.org/wiki/Differentiation_rules "Differentiation rules") [linearity](https://en.wikipedia.org/wiki/Linearity_of_differentiation "Linearity of differentiation") [Power](https://en.wikipedia.org/wiki/Power_rule "Power rule") [Sum](https://en.wikipedia.org/wiki/Sum_rule_in_differentiation "Sum rule in differentiation") [Chain](https://en.wikipedia.org/wiki/Chain_rule "Chain rule") [L'Hôpital's](https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule "L'Hôpital's rule") [Product](https://en.wikipedia.org/wiki/Product_rule "Product rule") [General Leibniz's rule](https://en.wikipedia.org/wiki/General_Leibniz_rule "General Leibniz rule") [Quotient](https://en.wikipedia.org/wiki/Quotient_rule "Quotient rule") Other techniques [Implicit differentiation](https://en.wikipedia.org/wiki/Implicit_differentiation "Implicit differentiation") [Inverse function rule](https://en.wikipedia.org/wiki/Inverse_function_rule "Inverse function rule") [Logarithmic derivative](https://en.wikipedia.org/wiki/Logarithmic_derivative "Logarithmic derivative") [Related rates](https://en.wikipedia.org/wiki/Related_rates "Related rates") [Stationary points](https://en.wikipedia.org/wiki/Stationary_point "Stationary point") [First derivative test](https://en.wikipedia.org/wiki/First_derivative_test "First derivative test") [Second derivative test](https://en.wikipedia.org/wiki/Second_derivative_test "Second derivative test") [Extreme value theorem](https://en.wikipedia.org/wiki/Extreme_value_theorem "Extreme value theorem") [Maximum and minimum](https://en.wikipedia.org/wiki/Maximum_and_minimum "Maximum and minimum") Further applications [Newton's method](https://en.wikipedia.org/wiki/Newton%27s_method "Newton's method") [Taylor's theorem](https://en.wikipedia.org/wiki/Taylor%27s_theorem "Taylor's theorem") [Differential equation](https://en.wikipedia.org/wiki/Differential_equation "Differential equation") [Ordinary differential equation](https://en.wikipedia.org/wiki/Ordinary_differential_equation "Ordinary differential equation") [Partial differential equation](https://en.wikipedia.org/wiki/Partial_differential_equation "Partial differential equation") [Stochastic differential equation](https://en.wikipedia.org/wiki/Stochastic_differential_equation "Stochastic differential equation") |
| [Integral calculus](https://en.wikipedia.org/wiki/Integral_calculus "Integral calculus") | [Antiderivative](https://en.wikipedia.org/wiki/Antiderivative "Antiderivative") [Arc length](https://en.wikipedia.org/wiki/Arc_length "Arc length") [Riemann integral](https://en.wikipedia.org/wiki/Riemann_integral "Riemann integral") [Basic properties](https://en.wikipedia.org/wiki/Integral#Properties "Integral") [Constant of integration](https://en.wikipedia.org/wiki/Constant_of_integration "Constant of integration") [Fundamental theorem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus "Fundamental theorem of calculus") [Differentiating under the integral sign](https://en.wikipedia.org/wiki/Leibniz_integral_rule "Leibniz integral rule") [Integration by parts](https://en.wikipedia.org/wiki/Integration_by_parts "Integration by parts") [Integration by substitution](https://en.wikipedia.org/wiki/Integration_by_substitution "Integration by substitution") [trigonometric](https://en.wikipedia.org/wiki/Trigonometric_substitution "Trigonometric substitution") [Euler](https://en.wikipedia.org/wiki/Euler_substitution "Euler substitution") [Tangent half-angle substitution](https://en.wikipedia.org/wiki/Tangent_half-angle_substitution "Tangent half-angle substitution") [Partial fractions in integration](https://en.wikipedia.org/wiki/Partial_fractions_in_integration "Partial fractions in integration") [Quadratic integral](https://en.wikipedia.org/wiki/Quadratic_integral "Quadratic integral") [Trapezoidal rule](https://en.wikipedia.org/wiki/Trapezoidal_rule "Trapezoidal rule") Volumes [Washer method](https://en.wikipedia.org/wiki/Disc_integration "Disc integration") [Shell method](https://en.wikipedia.org/wiki/Shell_integration "Shell integration") [Integral equation](https://en.wikipedia.org/wiki/Integral_equation "Integral equation") [Integro-differential equation](https://en.wikipedia.org/wiki/Integro-differential_equation "Integro-differential equation") |
| [Vector calculus](https://en.wikipedia.org/wiki/Vector_calculus "Vector calculus") | Derivatives [Curl](https://en.wikipedia.org/wiki/Curl_\(mathematics\) "Curl (mathematics)") [Directional derivative](https://en.wikipedia.org/wiki/Directional_derivative "Directional derivative") [Divergence](https://en.wikipedia.org/wiki/Divergence "Divergence") [Gradient](https://en.wikipedia.org/wiki/Gradient "Gradient") [Laplacian](https://en.wikipedia.org/wiki/Laplace_operator "Laplace operator") Basic theorems [Line integrals](https://en.wikipedia.org/wiki/Fundamental_Theorem_of_Line_Integrals "Fundamental Theorem of Line Integrals") [Green's](https://en.wikipedia.org/wiki/Green%27s_theorem "Green's theorem") [Stokes'](https://en.wikipedia.org/wiki/Stokes%27_theorem "Stokes' theorem") [Gauss'](https://en.wikipedia.org/wiki/Divergence_theorem "Divergence theorem") |
| [Multivariable calculus](https://en.wikipedia.org/wiki/Multivariable_calculus "Multivariable calculus") | [Divergence theorem](https://en.wikipedia.org/wiki/Divergence_theorem "Divergence theorem") [Geometric](https://en.wikipedia.org/wiki/Geometric_calculus "Geometric calculus") [Hessian matrix](https://en.wikipedia.org/wiki/Hessian_matrix "Hessian matrix") [Jacobian matrix and determinant](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant "Jacobian matrix and determinant") [Lagrange multiplier](https://en.wikipedia.org/wiki/Lagrange_multiplier "Lagrange multiplier") [Line integral](https://en.wikipedia.org/wiki/Line_integral "Line integral") [Matrix](https://en.wikipedia.org/wiki/Matrix_calculus "Matrix calculus") [Multiple integral](https://en.wikipedia.org/wiki/Multiple_integral "Multiple integral") [Partial derivative]() [Surface integral](https://en.wikipedia.org/wiki/Surface_integral "Surface integral") [Volume integral](https://en.wikipedia.org/wiki/Volume_integral "Volume integral") Advanced topics [Differential forms](https://en.wikipedia.org/wiki/Differential_form "Differential form") [Exterior derivative](https://en.wikipedia.org/wiki/Exterior_derivative "Exterior derivative") [Generalized Stokes' theorem](https://en.wikipedia.org/wiki/Generalized_Stokes%27_theorem "Generalized Stokes' theorem") [Tensor calculus](https://en.wikipedia.org/wiki/Tensor_calculus "Tensor calculus") |
| Sequences and series | [Arithmetico-geometric sequence](https://en.wikipedia.org/wiki/Arithmetico-geometric_sequence "Arithmetico-geometric sequence") Types of series [Alternating](https://en.wikipedia.org/wiki/Alternating_series "Alternating series") [Binomial](https://en.wikipedia.org/wiki/Binomial_series "Binomial series") [Fourier](https://en.wikipedia.org/wiki/Fourier_series "Fourier series") [Geometric](https://en.wikipedia.org/wiki/Geometric_series "Geometric series") [Harmonic](https://en.wikipedia.org/wiki/Harmonic_series_\(mathematics\) "Harmonic series (mathematics)") [Infinite](https://en.wikipedia.org/wiki/Infinite_series "Infinite series") [Power](https://en.wikipedia.org/wiki/Power_series "Power series") [Maclaurin](https://en.wikipedia.org/wiki/Maclaurin_series "Maclaurin series") [Taylor](https://en.wikipedia.org/wiki/Taylor_series "Taylor series") [Telescoping](https://en.wikipedia.org/wiki/Telescoping_series "Telescoping series") Tests of convergence [Abel's](https://en.wikipedia.org/wiki/Abel%27s_test "Abel's test") [Alternating series](https://en.wikipedia.org/wiki/Alternating_series_test "Alternating series test") [Cauchy condensation](https://en.wikipedia.org/wiki/Cauchy_condensation_test "Cauchy condensation test") [Direct comparison](https://en.wikipedia.org/wiki/Direct_comparison_test "Direct comparison test") [Dirichlet's](https://en.wikipedia.org/wiki/Dirichlet%27s_test "Dirichlet's test") [Integral](https://en.wikipedia.org/wiki/Integral_test_for_convergence "Integral test for convergence") [Limit comparison](https://en.wikipedia.org/wiki/Limit_comparison_test "Limit comparison test") [Ratio](https://en.wikipedia.org/wiki/Ratio_test "Ratio test") [Root](https://en.wikipedia.org/wiki/Root_test "Root test") [Term](https://en.wikipedia.org/wiki/Term_test "Term test") |
| Special functions and numbers | [Bernoulli numbers](https://en.wikipedia.org/wiki/Bernoulli_number "Bernoulli number") [e (mathematical constant)](https://en.wikipedia.org/wiki/E_\(mathematical_constant\) "E (mathematical constant)") [Exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") [Natural logarithm](https://en.wikipedia.org/wiki/Natural_logarithm "Natural logarithm") [Stirling's approximation](https://en.wikipedia.org/wiki/Stirling%27s_approximation "Stirling's approximation") |
| [History of calculus](https://en.wikipedia.org/wiki/History_of_calculus "History of calculus") | [Adequality](https://en.wikipedia.org/wiki/Adequality "Adequality") [Brook Taylor](https://en.wikipedia.org/wiki/Brook_Taylor "Brook Taylor") [Colin Maclaurin](https://en.wikipedia.org/wiki/Colin_Maclaurin "Colin Maclaurin") [Generality of algebra](https://en.wikipedia.org/wiki/Generality_of_algebra "Generality of algebra") [Gottfried Wilhelm Leibniz](https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz "Gottfried Wilhelm Leibniz") [Infinitesimal](https://en.wikipedia.org/wiki/Infinitesimal "Infinitesimal") [Infinitesimal calculus](https://en.wikipedia.org/wiki/Infinitesimal_calculus "Infinitesimal calculus") [Isaac Newton](https://en.wikipedia.org/wiki/Isaac_Newton "Isaac Newton") [Fluxion](https://en.wikipedia.org/wiki/Fluxion "Fluxion") [Law of Continuity](https://en.wikipedia.org/wiki/Law_of_Continuity "Law of Continuity") [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") *[Method of Fluxions](https://en.wikipedia.org/wiki/Method_of_Fluxions "Method of Fluxions")* *[The Method of Mechanical Theorems](https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems "The Method of Mechanical Theorems")* |
| Lists | |
| | |
| [Integrals](https://en.wikipedia.org/wiki/Lists_of_integrals "Lists of integrals") | [rational functions](https://en.wikipedia.org/wiki/List_of_integrals_of_rational_functions "List of integrals of rational functions") [irrational algebraic functions](https://en.wikipedia.org/wiki/List_of_integrals_of_irrational_algebraic_functions "List of integrals of irrational algebraic functions") [exponential functions](https://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions "List of integrals of exponential functions") [logarithmic functions](https://en.wikipedia.org/wiki/List_of_integrals_of_logarithmic_functions "List of integrals of logarithmic functions") [hyperbolic functions](https://en.wikipedia.org/wiki/List_of_integrals_of_hyperbolic_functions "List of integrals of hyperbolic functions") [inverse](https://en.wikipedia.org/wiki/List_of_integrals_of_inverse_hyperbolic_functions "List of integrals of inverse hyperbolic functions") [trigonometric functions](https://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions "List of integrals of trigonometric functions") [inverse](https://en.wikipedia.org/wiki/List_of_integrals_of_inverse_trigonometric_functions "List of integrals of inverse trigonometric functions") [Secant](https://en.wikipedia.org/wiki/Integral_of_the_secant_function "Integral of the secant function") [Secant cubed](https://en.wikipedia.org/wiki/Integral_of_secant_cubed "Integral of secant cubed") |
| [List of limits](https://en.wikipedia.org/wiki/List_of_limits "List of limits") [List of derivatives](https://en.wikipedia.org/wiki/Differentiation_rules "Differentiation rules") | |
| Miscellaneous topics | Complex calculus [Contour integral](https://en.wikipedia.org/wiki/Contour_integral "Contour integral") Differential geometry [Manifold](https://en.wikipedia.org/wiki/Manifold "Manifold") [Curvature](https://en.wikipedia.org/wiki/Curvature "Curvature") [of curves](https://en.wikipedia.org/wiki/Differential_geometry_of_curves "Differential geometry of curves") [of surfaces](https://en.wikipedia.org/wiki/Differential_geometry_of_surfaces "Differential geometry of surfaces") [Tensor](https://en.wikipedia.org/wiki/Tensor "Tensor") [Euler–Maclaurin formula](https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula "Euler–Maclaurin formula") [Gabriel's horn](https://en.wikipedia.org/wiki/Gabriel%27s_horn "Gabriel's horn") [Integration Bee](https://en.wikipedia.org/wiki/Integration_Bee "Integration Bee") [Proof that 22/7 exceeds π](https://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80 "Proof that 22/7 exceeds π") [Regiomontanus' angle maximization problem](https://en.wikipedia.org/wiki/Regiomontanus%27_angle_maximization_problem "Regiomontanus' angle maximization problem") [Steinmetz solid](https://en.wikipedia.org/wiki/Steinmetz_solid "Steinmetz solid") |
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Partial derivative
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| Readable Markdown | In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), a **partial derivative** of a [function of several variables](https://en.wikipedia.org/wiki/Function_\(mathematics\)#MULTIVARIATE_FUNCTION "Function (mathematics)") is its [derivative](https://en.wikipedia.org/wiki/Derivative "Derivative") with respect to one of those variables, with the others held constant (as opposed to the [total derivative](https://en.wikipedia.org/wiki/Total_derivative "Total derivative"), in which all variables are allowed to vary). Partial derivatives are used in [vector calculus](https://en.wikipedia.org/wiki/Vector_calculus "Vector calculus") and [differential geometry](https://en.wikipedia.org/wiki/Differential_geometry "Differential geometry").
The partial derivative of a function  with respect to the variable  (analogously for any other variable) is variously denoted by
, , , ,  , , or .
It is 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 [∂](https://en.wikipedia.org/wiki/%E2%88%82 "∂"). One of the first known uses of this symbol in mathematics is by [Marquis de Condorcet](https://en.wikipedia.org/wiki/Marquis_de_Condorcet "Marquis de Condorcet") from 1770,[\[1\]](https://en.wikipedia.org/wiki/Partial_derivative#cite_note-Cajori_History_V2-1) who used it for [partial differences](https://en.wikipedia.org/wiki/Partial_difference_equation "Partial difference equation"). The modern partial derivative notation was created by [Adrien-Marie Legendre](https://en.wikipedia.org/wiki/Adrien-Marie_Legendre "Adrien-Marie Legendre") (1786), although he later abandoned it; [Carl Gustav Jacob Jacobi](https://en.wikipedia.org/wiki/Carl_Gustav_Jacob_Jacobi "Carl Gustav Jacob Jacobi") reintroduced the symbol in 1841.[\[2\]](https://en.wikipedia.org/wiki/Partial_derivative#cite_note-miller_earliest-2)
Like ordinary derivatives, the partial derivative is defined as a [limit](https://en.wikipedia.org/wiki/Limit_of_a_function "Limit of a function"). Let U be an [open subset](https://en.wikipedia.org/wiki/Open_set "Open set") of  and  a function. The partial derivative of f at the point  with respect to the i\-th variable *x**i* is defined as
where  is the [unit vector](https://en.wikipedia.org/wiki/Unit_vector "Unit vector") of i\-th variable *x**i*. In fact, the last equality shows that the partial derivative is just the [directional derivative](https://en.wikipedia.org/wiki/Directional_derivative "Directional derivative") where the direction is the \-th [standard basis](https://en.wikipedia.org/wiki/Standard_basis "Standard basis") vector.
Even if all partial derivatives  exist at a given point a, the function need not be [continuous](https://en.wikipedia.org/wiki/Continuous_function "Continuous function") there. However, if all partial derivatives exist in a [neighborhood](https://en.wikipedia.org/wiki/Neighborhood_\(topology\) "Neighborhood (topology)") of a and are continuous there, then f is [totally differentiable](https://en.wikipedia.org/wiki/Total_derivative "Total derivative") 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 component-wise argument.
The partial derivative  is itself a function defined on U and can be partially-differentiated again. If the direction of derivative is *not* repeated, it is called a ***mixed partial derivative***. 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](https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Schwarz's_theorem "Symmetry of second derivatives"):
Further information: [∂](https://en.wikipedia.org/wiki/%E2%88%82 "∂")
For the following examples, let f be a function in x, y, and z.
First-order partial derivatives:
Second-order partial derivatives:
Second-order [mixed derivatives](https://en.wikipedia.org/wiki/Mixed_derivatives "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](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics"), the partial derivative of f with respect to x, holding y and z 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](https://en.wikipedia.org/wiki/Abuse_of_notation "Abuse of notation") 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 first variable.[\[3\]](https://en.wikipedia.org/wiki/Partial_derivative#cite_note-3)
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](https://en.wikipedia.org/wiki/Clairaut%27s_theorem_on_equality_of_mixed_partials "Clairaut's theorem on equality of mixed partials") implies that  as long as comparatively mild regularity conditions on f are satisfied.
An important example of a function of several variables is the case of a [scalar-valued function](https://en.wikipedia.org/wiki/Scalar-valued_function "Scalar-valued function")  on a domain in Euclidean space  (e.g., on  or ). In this case f has a partial derivative  with respect to each variable *x**j*. At the point a, these partial derivatives define the vector
This vector is called the *[gradient](https://en.wikipedia.org/wiki/Gradient "Gradient")* 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](https://en.wikipedia.org/wiki/Vector_field "Vector field").
A common [abuse of notation](https://en.wikipedia.org/wiki/Abuse_of_notation "Abuse of notation") is to define the [del operator](https://en.wikipedia.org/wiki/Del_operator "Del operator") (∇) as follows in three-dimensional [Euclidean space](https://en.wikipedia.org/wiki/Euclidean_space "Euclidean space")  with [unit vectors](https://en.wikipedia.org/wiki/Unit_vectors "Unit vectors") :
![{\\displaystyle \\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} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c70b5bce4676294a2be68361d333b3f44ce478)
Or, more generally, for n\-dimensional Euclidean space  with coordinates  and unit vectors :
![{\\displaystyle \\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4449e767ec489248b3a285415bf722bd550932c7)
## Directional derivative
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=4 "Edit section: Directional derivative")\]
Suppose that f is a function of more than one variable. For instance,
[](https://en.wikipedia.org/wiki/File:Partial_func_eg.svg)
A graph of *z* = *x*2 + *xy* + *y*2. For the partial derivative at (1, 1) that leaves y constant, the corresponding [tangent](https://en.wikipedia.org/wiki/Tangent "Tangent") line is parallel to the xz\-plane.
[](https://en.wikipedia.org/wiki/File:X2%2BX%2B1.svg)
A slice of the graph above showing the function in the xz\-plane at *y* = 1. The two axes are shown here with different scales. The slope of the tangent line is 3.
The [graph](https://en.wikipedia.org/wiki/Graph_of_a_function "Graph of a function") of this function defines a [surface](https://en.wikipedia.org/wiki/Surface_\(topology\) "Surface (topology)") in [Euclidean space](https://en.wikipedia.org/wiki/Euclidean_space "Euclidean space"). To every point on this surface, there are an infinite number of [tangent lines](https://en.wikipedia.org/wiki/Tangent_line "Tangent line"). Partial differentiation is the act of choosing one of these lines and finding its [slope](https://en.wikipedia.org/wiki/Slope "Slope"). 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](https://en.wikipedia.org/wiki/Derivative "Derivative") of the equation while assuming that y is a constant, we find that the slope of f at the point (*x*, *y*) is:
So at (1, 1), by substitution, the slope is 3. Therefore,
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:
In other words, every value of y defines a function, denoted *fy*, which is a function of one variable x.[\[6\]](https://en.wikipedia.org/wiki/Partial_derivative#cite_note-6) That is,
In this section the subscript notation *fy* 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 *fa* which traces a curve *x*2 + *ax* + *a*2 on the xz\-plane:
In this expression, a is a *constant*, not a *variable*, so *fa* 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](https://en.wikipedia.org/wiki/Partial_derivative_symbol "Partial derivative symbol")*; to distinguish it from the letter 'd', '∂' is sometimes pronounced "partial".
## Higher order partial derivatives
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=6 "Edit section: Higher order partial derivatives")\]
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):[\[7\]](https://en.wikipedia.org/wiki/Partial_derivative#cite_note-7): 316–318
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](https://en.wikipedia.org/wiki/Schwarz_theorem "Schwarz 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](https://en.wikipedia.org/wiki/Hessian_matrix "Hessian matrix") which is used in the [second order conditions](https://en.wikipedia.org/wiki/Second_order_condition "Second order condition") in [optimization](https://en.wikipedia.org/wiki/Optimization "Optimization") problems. The higher order partial derivatives can be obtained by successive differentiation
## Antiderivative analogue
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=7 "Edit section: Antiderivative analogue")\]
There is a concept for partial derivatives that is analogous to [antiderivatives](https://en.wikipedia.org/wiki/Antiderivative "Antiderivative") for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function.
Consider the example of
The so-called 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](https://en.wikipedia.org/wiki/Constant_of_integration "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 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 , 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](https://en.wikipedia.org/wiki/Gradient "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](https://en.wikipedia.org/wiki/Conservative_vector_field "Conservative vector field").
[](https://en.wikipedia.org/wiki/File:Cone_3d.png)
The volume of a cone depends on height and radius
The [volume](https://en.wikipedia.org/wiki/Volume "Volume") V of a [cone](https://en.wikipedia.org/wiki/Cone_\(geometry\) "Cone (geometry)") depends on the cone's [height](https://en.wikipedia.org/wiki/Height "Height") h and its [radius](https://en.wikipedia.org/wiki/Radius "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 h 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](https://en.wikipedia.org/wiki/Total_derivative "Total derivative") of V with respect to r and h are respectively
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](https://en.wikipedia.org/wiki/Gradient "Gradient") vector
Partial derivatives appear in any calculus-based [optimization](https://en.wikipedia.org/wiki/Optimization "Optimization") problem with more than one choice variable. For example, in [economics](https://en.wikipedia.org/wiki/Economics "Economics") a firm may wish to maximize [profit](https://en.wikipedia.org/wiki/Profit_\(economics\) "Profit (economics)") π(*x*, *y*) with respect to the choice of the quantities x and y of two different types of output. The [first order conditions](https://en.wikipedia.org/wiki/First_order_condition "First order condition") 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](https://en.wikipedia.org/wiki/System_of_equations "System of equations").
### Thermodynamics, quantum mechanics and mathematical physics
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_derivative&action=edit§ion=11 "Edit section: Thermodynamics, quantum mechanics and mathematical physics")\]
Partial derivatives appear in thermodynamic equations like [Gibbs-Duhem equation](https://en.wikipedia.org/wiki/Gibbs-Duhem_equation "Gibbs-Duhem equation"), in quantum mechanics as in [Schrödinger wave equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation"), as well as in other equations from [mathematical physics](https://en.wikipedia.org/wiki/Mathematical_physics "Mathematical physics"). The variables being held constant in partial derivatives here can be ratios of simple variables like [mole fractions](https://en.wikipedia.org/wiki/Mole_fraction "Mole fraction") *xi* in the following example involving the Gibbs energies in a ternary mixture system:
Express [mole fractions](https://en.wikipedia.org/wiki/Mole_fraction "Mole fraction") 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](https://en.wikipedia.org/wiki/Partial_differential_equation "Partial differential equation") like:
This equality can be rearranged to have differential quotient of mole fractions on one side.
Partial derivatives are key to target-aware image resizing algorithms. Widely known as [seam carving](https://en.wikipedia.org/wiki/Seam_carving "Seam carving"), these algorithms require each [pixel](https://en.wikipedia.org/wiki/Pixel "Pixel") in an image to be assigned a numerical 'energy' to describe their dissimilarity against orthogonal adjacent pixels. The [algorithm](https://en.wikipedia.org/wiki/Algorithm "Algorithm") then progressively removes rows or columns with the lowest energy. The formula established to determine a pixel's energy (magnitude of [gradient](https://en.wikipedia.org/wiki/Gradient "Gradient") at a pixel) depends heavily on the constructs of partial derivatives.
Partial derivatives play a prominent role in [economics](https://en.wikipedia.org/wiki/Economics "Economics"), in which most functions describing economic behaviour posit that the behaviour depends on more than one variable. For example, a societal [consumption function](https://en.wikipedia.org/wiki/Consumption_function "Consumption function") may describe the amount spent on consumer goods as depending on both income and wealth; the [marginal propensity to consume](https://en.wikipedia.org/wiki/Marginal_propensity_to_consume "Marginal propensity to consume") is then the partial derivative of the consumption function with respect to income.
- [d'Alembert operator](https://en.wikipedia.org/wiki/D%27Alembert_operator "D'Alembert operator")
- [Chain rule](https://en.wikipedia.org/wiki/Chain_rule "Chain rule")
- [Curl (mathematics)](https://en.wikipedia.org/wiki/Curl_\(mathematics\) "Curl (mathematics)")
- [Divergence](https://en.wikipedia.org/wiki/Divergence "Divergence")
- [Exterior derivative](https://en.wikipedia.org/wiki/Exterior_derivative "Exterior derivative")
- [Iterated integral](https://en.wikipedia.org/wiki/Iterated_integral "Iterated integral")
- [Jacobian matrix and determinant](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant "Jacobian matrix and determinant")
- [Laplace operator](https://en.wikipedia.org/wiki/Laplace_operator "Laplace operator")
- [Multivariable calculus](https://en.wikipedia.org/wiki/Multivariable_calculus "Multivariable calculus")
- [Symmetry of second derivatives](https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives "Symmetry of second derivatives")
- [Triple product rule](https://en.wikipedia.org/wiki/Triple_product_rule "Triple product rule"), also known as the cyclic chain rule.
1. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-Cajori_History_V2_1-0)**
Cajori, Florian (1952), [*A History of Mathematical Notations*](https://archive.org/details/AHistoryOfMathematicalNotationVolII/page/n153/mode/2up), vol. 2 (3 ed.), The Open Court Publishing Company, 596
2. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-miller_earliest_2-0)**
Miller, Jeff (n.d.). ["Earliest Uses of Symbols of Calculus"](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/calculus/). In O'Connor, John J.; [Robertson, Edmund F.](https://en.wikipedia.org/wiki/Edmund_F._Robertson "Edmund F. Robertson") (eds.). *[MacTutor History of Mathematics archive](https://en.wikipedia.org/wiki/MacTutor_History_of_Mathematics_archive "MacTutor History of Mathematics archive")*. [University of St Andrews](https://en.wikipedia.org/wiki/University_of_St_Andrews "University of St Andrews"). Retrieved 2023-06-15.
3. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-3)**
Spivak, M. (1965). [*Calculus on Manifolds*](https://archive.org/details/SpivakM.CalculusOnManifoldsPerseus2006Reprint). New York: W. A. Benjamin. p. 44. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9780805390216](https://en.wikipedia.org/wiki/Special:BookSources/9780805390216 "Special:BookSources/9780805390216")
.
4. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-4)**
R. Wrede; M.R. Spiegel (2010). *Advanced Calculus* (3rd ed.). Schaum's Outline Series. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-162366-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-162366-7 "Special:BookSources/978-0-07-162366-7")
.
5. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-5)** The applicability extends to functions over spaces without a [metric](https://en.wikipedia.org/wiki/Metric_\(mathematics\) "Metric (mathematics)") and to [differentiable manifolds](https://en.wikipedia.org/wiki/Differentiable_manifold "Differentiable manifold"), such as in [general relativity](https://en.wikipedia.org/wiki/General_relativity "General relativity").
6. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-6)** This can also be expressed as the [adjointness](https://en.wikipedia.org/wiki/Adjoint_functors "Adjoint functors") between the [product space](https://en.wikipedia.org/wiki/Product_topology "Product topology") and [function space](https://en.wikipedia.org/wiki/Function_space "Function space") constructions.
7. **[^](https://en.wikipedia.org/wiki/Partial_derivative#cite_ref-7)**
[Chiang, Alpha C.](https://en.wikipedia.org/wiki/Alpha_Chiang "Alpha Chiang") (1984). *Fundamental Methods of Mathematical Economics* (3rd ed.). McGraw-Hill.
- ["Partial derivative"](https://www.encyclopediaofmath.org/index.php?title=Partial_derivative), *[Encyclopedia of Mathematics](https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics "Encyclopedia of Mathematics")*, [EMS Press](https://en.wikipedia.org/wiki/European_Mathematical_Society "European Mathematical Society"), 2001 \[1994\]
- [Partial Derivatives](http://mathworld.wolfram.com/PartialDerivative.html) at [MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld") |
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