Lihat sumber Turunan parsial

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Dalam [[matematika]], '''turunan parsial''' sebuah [[fungsi matematika]] [[peubah]] banyak adalah turunannya terhadap salah satu peubah (variabel) dengan peubah lainnya dipertahankan (konstan). Ini dibedakan dengan [[turunan total]], yang membolehkan semua variabelnya untuk berubah. Turunan parsial berguna dalam bidang [[kalkulus vektor]] dan [[geometri diferensial]]

Turunan parsial sebuah fungsi ''f'' terhadap variabel ''x'' dituliskan oleh berbagai sumber rujukan sebagai

: <math>f^\prime_x,\  f_x,\  \partial_x f, \text{ atau}  \frac{\partial f}{\partial x}.</math>

Lambang turunan parsial ''[[∂]]'' adalah huruf bundar, diturunkan namun berbeda dengan huruf [[Yunani]] delta, dan dibedakan dengan notasi turunan total ''d'' (dan dari huruf [[ð]])

== Pengantar ==
Jika {{math|''f''}} adalah fungsi lebih dari satu variabel. Misalnya,
:<math>z = f(x,y) = x^2 + xy + y^2.</math>

{{multiple image
 | align  = right
 | direction = vertical
 | width  = 250

 | image1 = Partial func eg.svg
 | caption1 = Grafik pada {{math|''z'' {{=}} ''x''<sup>2</sup> + ''xy'' + ''y''<sup>2</sup>}}. Untuk turunan parsial pada {{math|(''1'',''1'')}} yang meninggalkan {{math|''y''}} konstan, garis singgung terkait sejajar dengan bidang {{math|''xz''}}.

 | image2 = X2+X+1.svg
 | caption2 = Sepotong grafik di atas menunjukkan fungsi pada bidang {{math|''xz''}} pada {{math|''y'' {{=}} 1}}. Perhatikan bahwa kedua sumbu ditampilkan di sini dengan skala yang berbeda. Kemiringan garis singgung adalah 3.
}}

grafik dari fungsi tersebut merumuskan permukaan pada [[Ruang Euclides]]. Untuk setiap titik pada permukaan ini terdapat jumlah garis pinggir tidak terbatas. Antiturunan parsial salah satu garis yang ditemukannya kemiringan. Biasanya, garis yang paling terkenal adalah garis yang sejajar dengan <math>xz</math>, dan yang sejajar dengan {{math|''yz''}}.

Dengan cara mencari turunan dari persamaan sambil mengasumsikan <math>y</math> adalah konstan, kami menemukan bahwa kemiringan ''<math>f</math>'' pada intinya <math>(x, y)</math> adalah, sebagai berikut:

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

Jadi <math>(1, 1)</math>, dengan substitusi, kemiringan adalah 3. Oleh karena itu,

: <math>\frac{\partial z}{\partial x} = 3</math>

== Definisi ==
<!--=== Definisi dasar ===
Fungsi pada {{math|''f''}} dapat diartikan kembali sebagai keluarga fungsi dari satu variabel yang diindeks oleh variabel lainnya, yaitu:

: <math>f(x,y) = f_y(x) = x^2 + xy + y^2.</math>

Dengan kata lain, setiap nilai {{math|''y''}} mendefinisikan fungsi, dilambangkan dengan nilai {{math|''f''<sub>y</sub>}} merupakan salah satu fungsi dari satu variabel {{math|''x''}}.{{efn|This can also be expressed as the [[adjoint functors|adjointness]] between the [[product topology|product space]] and [[function space]] constructions.}} Yaitu,

: <math>f_y(x) = x^2 + xy + y^2.</math>

In this section the subscript notation ''f<sub>y</sub>'' 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<sub>a</sub>'' which traces a curve ''x''<sup>2</sup> + ''ax'' + ''a''<sup>2</sup> on the <math>xz</math>-plane:

: <math>f_a(x) = x^2 + ax + a^2. </math>

In this expression, ''a'' is a ''constant'', not a ''variable'', so ''f<sub>a</sub>'' is a function of only one real variable, that being ''x''. Consequently, the definition of the derivative for a function of one variable applies:

: <math>f_a'(x) = 2x + a.</math>

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:

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

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".

In general, the '''partial derivative''' of an [[Arity|n-ary]] function ''f''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) in the direction ''x<sub>i</sub>'' at the point (''a''<sub>1</sub>, ..., ''a<sub>n</sub>'') is defined to be:

: <math>\frac{\partial f}{\partial x_i}(a_1, \ldots, a_n) = \lim_{h \to 0}\frac{f(a_1, \ldots, a_i+h,\ldots,a_n) - f(a_1,\ldots, a_i, \dots,a_n)}{h}.</math>

In the above difference quotient, all the variables except ''x<sub>i</sub>'' are held fixed. That choice of fixed values determines a function of one variable

:<math>f_{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n}(x_i) = f(a_1,\ldots,a_{i-1},x_i,a_{i+1},\ldots,a_n),</math>

and by definition,

:<math>\frac{df_{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n}}{dx_i}(a_i) = \frac{\partial f}{\partial x_i}(a_1,\ldots,a_n).</math>

In other words, the different choices of ''a'' index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.

An important example of a function of several variables is the case of a [[scalar-valued function]] ''f''(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'') on a domain in Euclidean space <math>\R^n</math> (e.g., on <math>\R^2</math> or <math>\R^3</math>). In this case ''f'' has a partial derivative ∂''f''/∂''x<sub>j</sub>'' with respect to each variable ''x''<sub>''j''</sub>. At the point ''a'', these partial derivatives define the vector

: <math>\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \ldots, \frac{\partial f}{\partial x_n}(a)\right).</math>

This vector is called the '''[[gradient]]''' of ''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]] <math>\R^3</math> with [[unit vectors]] <math>\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}</math>:

: <math>\nabla = \left[{\frac{\partial}{\partial x}} \right] \hat{\mathbf{i}} + \left[{\frac{\partial}{\partial y}} \right] \hat{\mathbf{j}} + \left[{\frac{\partial}{\partial z}}\right] \hat{\mathbf{k}}</math>

Or, more generally, for ''n''-dimensional Euclidean space <math>\R^n</math> with coordinates <math>x_1, \ldots, x_n</math> and unit vectors <math>\hat{\mathbf{e}}_1, \ldots, \hat{\mathbf{e}}_n</math>:

: <math>\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 + \ldots + \left[\frac{\partial}{\partial x_n} \right] \hat{\mathbf{e}}_n</math>

===Formal definition===
Like ordinary derivatives, the partial derivative is defined as a [[limit of a function|limit]]. Let ''U'' be an [[open set|open subset]] of <math>\R^n</math> and <math>f:U\to\R</math> a function. The partial derivative of ''f'' at the point <math>\mathbf{a}=(a_1, \ldots, a_n) \in U</math> with respect to the ''i''-th variable ''x''<sub>''i''</sub> is defined as

:<math>\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}</math>

Even if all partial derivatives ∂''f''/∂''x''<sub>''i''</sub>(''a'') exist at a given point ''a'', the function need not be [[continuous function|continuous]] there. However, if all partial derivatives exist in a [[neighborhood (topology)|neighborhood]] of ''a'' and are continuous there, then ''f'' is [[total derivative|totally differentiable]] in that neighborhood and the total derivative is continuous. In this case, it is said that ''f'' is a C<sup>1</sup> function. This can be used to generalize for vector valued functions, <math>f:U \to \R^m,</math> by carefully using a componentwise argument.

The partial derivative <math>\frac{\partial f}{\partial x}</math> can be seen as another function defined on ''U'' and can again be partially differentiated. If all mixed second order partial derivatives are continuous at a point (or on a set), ''f'' is termed a C<sup>2</sup> function at that point (or on that set); in this case, the partial derivatives can be exchanged by [[Symmetry of second derivatives#Clairaut.27s theorem|Clairaut's theorem]]:

:<math>\frac{\partial^2f}{\partial x_i \partial x_j} = \frac{\partial^2f} {\partial x_j \partial x_i}.</math>-->

== Contoh ==
* Tentukan turunan kedua dari <math>z = f(x,y) = x^2 + xy + y^2</math>!
: <math>f(x,y) = x^2 + xy + y^2</math>

; Turunan pertama
: <math>f(x,y) = x^2 + xy + y^2</math>
: <math>f_x(x,y) = 2x + y</math>
: <math>f_y(x,y) = x + 2y</math>

; Turunan kedua
: <math>f_x(x,y) = 2x + y</math>
: <math>f_{xx}(x,y) = 2</math>
: <math>f_{xy}(x,y) = 1</math>
: <math>f_y(x,y) = x + 2y</math>
: <math>f_{yy}(x,y) = 2</math>
: <math>f_{yx}(x,y) = 1</math>

* Tentukan turunan kedua dari <math>z = f(x,y) = x^2 + xy + y^3</math>!
: <math>f(x,y) = x^2 + xy + y^3</math>

; Turunan pertama
: <math>f(x,y) = x^2 + xy + y^3</math>
: <math>f_x(x,y) = 2x + y</math>
: <math>f_y(x,y) = x + 3y^2</math>

; Turunan kedua
: <math>f_x(x,y) = 2x + y</math>
: <math>f_{xx}(x,y) = 2</math>
: <math>f_{xy}(x,y) = 1</math>
: <math>f_y(x,y) = x + 3y^2</math>
: <math>f_{yy}(x,y) = 6y</math>
: <math>f_{yx}(x,y) = 1</math>

<!--===Geometry===
[[Image:Cone 3d.png|thumb|The volume of a cone depends on height and radius]]
The [[volume]] ''V'' of a [[cone (geometry)|cone]] depends on the cone's [[height]] ''h'' and its [[radius]] ''r'' according to the formula

:<math>V(r, h) = \frac{\pi r^2 h}{3}.</math>

The partial derivative of ''V'' with respect to ''r'' is

:<math>\frac{ \partial V}{\partial r} = \frac{ 2 \pi r h}{3},</math>

which represents the rate with which a cone's volume changes if its radius is varied and its height is kept constant. The partial derivative with respect to <math>h</math> equals <math>\frac{\pi r^2}{3},</math> which represents the rate with which the volume changes if its height is varied and its radius is kept constant.

By contrast, the [[total derivative|''total'' derivative]] of ''V'' with respect to ''r'' and ''h'' are respectively

:<math>\frac{dV}{dr} = \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r} + \overbrace{\frac{\pi r^2}{3}}^\frac{ \partial V}{\partial h}\frac{dh}{dr}</math>

and

:<math>\frac{dV}{dh} = \overbrace{\frac{\pi r^2}{3}}^\frac{\partial V}{\partial h} + \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r}\frac{dr}{dh}</math>

The difference between the total and partial derivative is the elimination of indirect dependencies between variables in partial derivatives.

If (for some arbitrary reason) the cone's proportions have to stay the same, and the height and radius are in a fixed ratio ''k'',

:<math>k = \frac{h}{r} = \frac{dh}{dr}.</math>

This gives the total derivative with respect to ''r'':

:<math>\frac{dV}{dr} = \frac{2 \pi r h}{3} + \frac{\pi r^2}{3}k</math>

which simplifies to:

:<math>\frac{dV}{dr} = k \pi r^2</math>

Similarly, the total derivative with respect to ''h'' is:

:<math>\frac{dV}{dh} = \pi r^2</math>

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

:<math>\nabla V = \left(\frac{\partial V}{\partial r},\frac{\partial V}{\partial h}\right) = \left(\frac{2}{3}\pi rh, \frac{1}{3}\pi r^2\right)</math>.

===Optimization===

Partial derivatives appear in any calculus-based [[optimization]] problem with more than one choice variable. For example, in [[economics]] a firm may wish to maximize [[profit (economics)|profit]] π(''x'', ''y'') with respect to the choice of the quantities ''x'' and ''y'' of two different types of output. The [[first order condition]]s for this optimization are π<sub>''x''</sub> = 0 = π<sub>''y''</sub>. Since both partial derivatives π<sub>''x''</sub> and π<sub>''y''</sub> will generally themselves be functions of both arguments ''x'' and ''y'', these two first order conditions form a [[System of equations|system of two equations in two unknowns]].

===Thermodynamics, quantum mechanics and mathematical physics===

Partial derivatives appear in thermodynamic equations like [[Gibbs-Duhem equation]], in quantum mechanics as Schrodinger wave equation as well in other equations from [[mathematical physics]]. Here the variables being held constant in partial derivatives can be ratio of simple variables like [[mole fraction]]s ''x<sub>i</sub>'' in the following example involving the Gibbs energies in a ternary mixture system:

:<math>\bar{G_2}= G + (1-x_2) \left(\frac{{\partial G}}{{\partial x_2}}\right)_{\frac{x_1}{x_3}} </math>

Express [[mole fraction]]s of a component as functions of other components' mole fraction and binary mole ratios:

:<math>x_1 = \frac{1-x_2}{1+\frac{x_3}{x_1}}</math>
:<math>x_3 = \frac{1-x_2}{1+\frac{x_1}{x_3}}</math>

Differential quotients can be formed at constant ratios like those above:

:<math>\left(\frac{\partial x_1}{\partial x_2}\right)_{\frac{x_1}{x_3}} = - \frac{x_1}{1-x_2}</math>
:<math>\left(\frac{\partial x_3}{\partial x_2}\right)_{\frac{x_1}{x_3}} = - \frac{x_3}{1-x_2}</math>

Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems:

:<math>X = \frac{x_3}{x_1 + x_3}</math>
:<math>Y = \frac{x_3}{x_2 + x_3}</math>
:<math>Z = \frac{x_2}{x_1 + x_2}</math>

which can be used for solving [[partial differential equation]]s like:

:<math>\left(\frac{\partial \mu_2}{\partial n_1}\right)_{n_2, n_3} = \left(\frac{\partial \mu_1}{\partial n_2}\right)_{n_1, n_3}</math>

This equality can be rearranged to have differential quotient of mole fractions on one side.

===Image resizing===

Partial derivatives are key to target-aware image resizing algorithms. Widely known as [[seam carving]], these algorithms require each [[pixel]] in an image to be assigned a numerical 'energy' to describe their dissimilarity against orthogonal adjacent pixels. The [[algorithm]] then progressively removes rows or columns with the lowest energy. The formula established to determine a pixel's energy (magnitude of [[gradient]] at a pixel) depends heavily on the constructs of partial derivatives.

===Economics===

Partial derivatives play a prominent role in [[economics]], in which most functions describing economic behaviour posit that the behaviour depends on more than one variable. For example, a societal [[consumption function]] may describe the amount spent on consumer goods as depending on both income and wealth; the [[marginal propensity to consume]] is then the partial derivative of the consumption function with respect to income.-->

== Notasi ==
{{Lihat|∂}}
<!--For the following examples, let <math>f</math> be a function in <math>x, y</math> and <math>z</math>.

First-order partial derivatives:

:<math>\frac{ \partial f}{ \partial x} = f_x = \partial_x f.</math>

Second-order partial derivatives:

:<math>\frac{ \partial^2 f}{ \partial x^2} = f_{xx} = \partial_{xx} f = \partial_x^2 f.</math>

Second-order [[mixed derivatives]]:

:<math>\frac{\partial^2 f}{\partial y \,\partial x} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = (f_{x})_{y} = f_{xy} = \partial_{yx} f = \partial_y \partial_x f.</math>

Higher-order partial and mixed derivatives:

:<math>\frac{\partial^{i+j+k} f}{\partial x^i \partial y^j \partial z^k } = f^{(i, j, k)} = \partial_x^i \partial_y^j \partial_z^k f.</math>

When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as [[statistical mechanics]], the partial derivative of <math>f</math> with respect to <math>x</math>, holding <math>y</math> and <math>z</math> constant, is often expressed as

:<math>\left( \frac{\partial f}{\partial x} \right)_{y,z}.</math>

Conventionally, for clarity and simplicity of notation, the partial derivative ''function'' and the ''value'' of the function at a specific point are [[Abuse of notation|conflated]] by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like

:<math>\frac{\partial f(x,y,z)}{\partial x}</math>

is used for the function, while

:<math>\frac{\partial f(u,v,w)}{\partial u}</math>

might be used for the value of the function at the point <math>(x,y,z)=(u,v,w)</math>. However, this convention breaks down when we want to evaluate the partial derivative at a point like <math>(x,y,z)=(17, u+v, v^2)</math>. In such a case, evaluation of the function must be expressed in an unwieldy manner as

:<math>\frac{\partial f(x,y,z)}{\partial x}(17, u+v, v^2)</math>

or

:<math>\left. \frac{\partial f(x,y,z)}{\partial x}\right |_{(x,y,z)=(17, u+v, v^2)}</math>

in order to use the Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with <math>D_i</math> as the partial derivative symbol with respect to the ''i''th variable. For instance, one would write <math>D_1 f(17, u+v, v^2)</math> for the example described above, while the expression <math>D_1 f</math> represents the partial derivative ''function'' with respect to the 1st variable.<ref>{{Cite book| url=https://archive.org/details/SpivakM.CalculusOnManifoldsPerseus2006Reprint| title=Calculus on Manifolds| last=Spivak| first=M.| publisher=W. A. Benjamin, Inc.|year=1965|isbn=9780805390216|location=New York|pages=44}}</ref>

For higher order partial derivatives, the partial derivative (function) of <math>D_i f</math> with respect to the ''j''th variable is denoted <math>D_j(D_i f)=D_{i,j} f</math>. That is, <math>D_j\circ D_i =D_{i,j}</math>, 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 on equality of mixed partials|Clairaut's theorem]] implies that <math>D_{i,j}=D_{j,i}</math> as long as comparatively mild regularity conditions on ''f'' are satisfied.-->

== Analog [[Antiturunan]] ==
<!--There is a concept for partial derivatives that is analogous to [[antiderivative]]s for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function.

Consider the example of

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

The "partial" integral can be taken with respect to ''x'' (treating ''y'' as constant, in a similar manner to partial differentiation):

:<math>z = \int \frac{\partial z}{\partial x} \,dx = x^2 + xy + g(y)</math>

Here, the [[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 <math>x</math> 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 <math>x^2 + xy + g(y)</math>, 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 <math>2x + y</math>.

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 vector field|conservative]].-->

== Antiturunan parsial dengan urutan lebih tinggi ==

<!--Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the function <math>f(x, y, ...)</math> the "own" second partial derivative with respect to ''x'' is simply the partial derivative of the partial derivative (both with respect to ''x''):<ref>[[Alpha Chiang|Chiang, Alpha C.]] ''Fundamental Methods of Mathematical Economics'', McGraw-Hill, third edition, 1984.</ref>{{rp|316–318}}

:<math>\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}.</math>

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

:<math>\frac{{\partial ^2 f}}{{\partial y\, \partial x}} \equiv \partial \frac{{\partial f / \partial x}}{{\partial y}} \equiv \frac{{\partial f_x}}{{\partial y}} \equiv f_{{xy}}.</math>

[[Schwarz theorem|Schwarz's theorem]] states that if the second derivatives are continuous the expression for the cross partial derivative is unaffected by which variable the partial derivative is taken with respect to first and which is taken second. That is,

:<math>\frac {{\partial ^2 f}}{{\partial x\, \partial y}} = \frac{{\partial ^2 f}}{{\partial y\, \partial x}}</math>

or equivalently <math>f_{{xy}}=f_{{yx}}.</math>

Own and cross partial derivatives appear in the [[Hessian matrix]] which is used in the [[second order condition]]s in [[optimization]] problems.-->

== Lihat pula ==
{{div col|colwidth=27em}}
*[[Operator d'Alembertian]]
*[[Curl (matematika)]]
{{div col end}}

== Catatan ==
{{Notelist}}

== Referensi ==
{{Reflist}}

== Pranala luar ==
* {{Springer |title = Partial derivative |id = p/p071620}}
* [http://mathworld.wolfram.com/PartialDerivative.html Partial Derivatives] at [[MathWorld]]

[[Kategori:Kalkulus multivariabel]]
[[Kategori:Operasi Differential]]


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