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In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.

The partial derivative of a function f(x,y,…)displaystyle f(x,y,dots ) with respect to the variable xdisplaystyle x is variously denoted by

fx′,fx,∂xf, Dxf,D1f,∂∂xf, or ∂f∂x.displaystyle f'_x,f_x,partial _xf, D_xf,D_1f,frac partial partial xf,text or frac partial fpartial x.

Sometimes, for z=f(x,y,…),displaystyle z=f(x,y,ldots ), the partial derivative of zdisplaystyle z with respect to xdisplaystyle x is denoted as ∂z∂x.displaystyle tfrac partial zpartial 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:

fx(x,y,…),∂f∂x(x,y,…).displaystyle f_x(x,y,ldots ),frac partial fpartial x(x,y,ldots ).

The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786), though he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol again in 1841.^[1]

Introduction

Suppose that f is a function of more than one variable. For instance,

z=f(x,y)=x2+xy+y2.displaystyle z=f(x,y)=x^2+xy+y^2.

A graph of z = x² + xy + y². 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. Note that 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 xzdisplaystyle 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)displaystyle P(1,1) and parallel to the xzdisplaystyle xz-plane, we treat ydisplaystyle 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=1displaystyle y=1 . By finding the derivative of the equation while assuming that ydisplaystyle y is a constant, we find that the slope of fdisplaystyle f at the point (x,y)displaystyle (x,y) is:

∂z∂x=2x+y.displaystyle frac partial zpartial x=2x+y.

So at (1,1)displaystyle (1,1), by substitution, the slope is 3. Therefore,

∂z∂x=3displaystyle frac partial zpartial x=3

at the point (1,1)displaystyle (1,1). That is, the partial derivative of zdisplaystyle z with respect to xdisplaystyle x at (1,1)displaystyle (1,1) is 3, as shown in the graph.

Definition

Basic definition

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

f(x,y)=fy(x)=x2+xy+y2.displaystyle f(x,y)=f_y(x)=x^2+xy+y^2.

In other words, every value of y defines a function, denoted f_y , which is a function of one variable x.^[a] That is,

fy(x)=x2+xy+y2.displaystyle f_y(x)=x^2+xy+y^2.

Note that 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 x² + ax + a² on the xzdisplaystyle xz-plane:

fa(x)=x2+ax+a2.displaystyle f_a(x)=x^2+ax+a^2.

In this expression, a is a constant, not a variable, so f_a is a function of only one real variable, that being x. Consequently, the definition of the derivative for a function of one variable applies:

fa′(x)=2x+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)=2x+y.displaystyle frac partial fpartial 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. To distinguish it from the letter d, ∂ is sometimes pronounced "tho" or "partial".

In general, the partial derivative of an n-ary function f(x₁, ..., x_n) in the direction x_i at the point (a₁, ..., a_n) is defined to be:

∂f∂xi(a1,…,an)=limh→0f(a1,…,ai+h,…,an)−f(a1,…,ai,…,an)h.displaystyle frac partial fpartial x_i(a_1,ldots ,a_n)=lim _hto 0frac f(a_1,ldots ,a_i+h,ldots ,a_n)-f(a_1,ldots ,a_i,dots ,a_n)h.

In the above difference quotient, all the variables except x_i are held fixed. That choice of fixed values determines a function of one variable

fa1,…,ai−1,ai+1,…,an(xi)=f(a1,…,ai−1,xi,ai+1,…,an),displaystyle 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),

and by definition,

dfa1,…,ai−1,ai+1,…,andxi(ai)=∂f∂xi(a1,…,an).displaystyle frac df_a_1,ldots ,a_i-1,a_i+1,ldots ,a_ndx_i(a_i)=frac partial fpartial x_i(a_1,ldots ,a_n).

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(x₁, ..., x_n) on a domain in Euclidean space Rndisplaystyle mathbb R ^n (e.g., on R2displaystyle mathbb R ^2 or R3displaystyle mathbb R ^3). In this case f has a partial derivative ∂f/∂x_j with respect to each variable x_j. At the point a, these partial derivatives define the vector

∇f(a)=(∂f∂x1(a),…,∂f∂xn(a)).displaystyle nabla f(a)=left(frac partial fpartial x_1(a),ldots ,frac partial fpartial x_n(a)right).

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 R3displaystyle mathbb R ^3 with 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 xright]hat mathbf i +left[frac partial partial yright]hat mathbf j +left[frac partial partial zright]hat mathbf k

Or, more generally, for n-dimensional Euclidean space Rndisplaystyle mathbb R ^n with coordinates x1,…,xndisplaystyle x_1,ldots ,x_n and unit vectors e^1,…,e^ndisplaystyle hat mathbf e _1,ldots ,hat mathbf e _n:

∇=∑j=1n[∂∂xj]e^j=[∂∂x1]e^1+[∂∂x2]e^2+…+[∂∂xn]e^ndisplaystyle nabla =sum _j=1^nleft[frac partial partial x_jright]hat mathbf e _j=left[frac partial partial x_1right]hat mathbf e _1+left[frac partial partial x_2right]hat mathbf e _2+ldots +left[frac partial partial x_nright]hat mathbf e _n

Formal definition

Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of Rndisplaystyle mathbb R ^n and f:U→Rdisplaystyle f:Uto mathbb R a function. The partial derivative of f at the point a=(a1,…,an)∈Udisplaystyle mathbf a =(a_1,ldots ,a_n)in U with respect to the i-th variable x_i is defined as

∂∂xif(a)=limh→0f(a1,…,ai−1,ai+h,ai+1,…,an)−f(a1,…,ai,…,an)hdisplaystyle frac partial partial x_if(mathbf a )=lim _hto 0frac 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

Even if all partial derivatives ∂f/∂x_i(a) 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 C¹ function. This can be used to generalize for vector valued functions, f:U→Rm,displaystyle f:Uto mathbb R ^m, by carefully using a componentwise argument.

The partial derivative ∂f∂xdisplaystyle frac partial fpartial x 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 C² function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:

∂2f∂xi∂xj=∂2f∂xj∂xi.displaystyle frac partial ^2fpartial x_ipartial x_j=frac partial ^2fpartial x_jpartial x_i.

Examples

Geometry

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

V(r,h)=πr2h3.displaystyle V(r,h)=frac pi r^2h3.

The partial derivative of V with respect to r is

∂V∂r=2πrh3,displaystyle frac partial Vpartial r=frac 2pi rh3,

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 hdisplaystyle h equals πr23,displaystyle frac pi r^23, 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

dVdr=2πrh3⏞∂V∂r+πr23⏞∂V∂hdhdrdisplaystyle frac dVdr=overbrace frac 2pi rh3 ^frac partial Vpartial r+overbrace frac pi r^23 ^frac partial Vpartial hfrac dhdr

and

dVdh=πr23⏞∂V∂h+2πrh3⏞∂V∂rdrdhdisplaystyle frac dVdh=overbrace frac pi r^23 ^frac partial Vpartial h+overbrace frac 2pi rh3 ^frac partial Vpartial rfrac drdh

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=hr=dhdr.displaystyle k=frac hr=frac dhdr.

This gives the total derivative with respect to r:

dVdr=2πrh3+πr23kdisplaystyle frac dVdr=frac 2pi rh3+frac pi r^23k

which simplifies to:

dVdr=kπr2displaystyle frac dVdr=kpi r^2

Similarly, the total derivative with respect to h is:

dVdh=πr2displaystyle frac dVdh=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 vector

∇V=(∂V∂r,∂V∂h)=(23πrh,13πr2)displaystyle nabla V=left(frac partial Vpartial r,frac partial Vpartial hright)=left(frac 23pi rh,frac 13pi r^2right).

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 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 and mathematical physics

Partial derivatives appear in thermodynamic equations like Gibbs-Duhem 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 fractions x_i in the following example involving the Gibbs energies in a ternary mixture system:

G2¯=G+(1−x2)(∂G∂x2)x1x3displaystyle bar G_2=G+(1-x_2)left(frac partial Gpartial x_2right)_frac x_1x_3

Express mole fractions of a component as functions of other components mole fraction and binary mole ratios:

x1=1−x21+x3x1displaystyle x_1=frac 1-x_21+frac x_3x_1

x3=1−x21+x1x3displaystyle x_3=frac 1-x_21+frac x_1x_3

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

(∂x1∂x2)x1x3=−x11−x2displaystyle left(frac partial x_1partial x_2right)_frac x_1x_3=-frac x_11-x_2

(∂x3∂x2)x1x3=−x31−x2displaystyle left(frac partial x_3partial x_2right)_frac x_1x_3=-frac x_31-x_2

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

X=x3x1+x3displaystyle X=frac x_3x_1+x_3

Y=x3x2+x3displaystyle Y=frac x_3x_2+x_3

Z=x2x1+x2displaystyle Z=frac x_2x_1+x_2

which can be used for solving partial differential equations like:

(∂μ2∂n1)n2,n3=(∂μ1∂n2)n1,n3displaystyle left(frac partial mu _2partial n_1right)_n_2,n_3=left(frac partial mu _1partial n_2right)_n_1,n_3

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.

Notation

For the following examples, let fdisplaystyle f be a function in x,ydisplaystyle x,y and zdisplaystyle z.

First-order partial derivatives:

∂f∂x=fx=∂xf.displaystyle frac partial fpartial x=f_x=partial _xf.

Second-order partial derivatives:

∂2f∂x2=fxx=∂xxf=∂x2f.displaystyle frac partial ^2fpartial x^2=f_xx=partial _xxf=partial _x^2f.

Second-order mixed derivatives:

∂2f∂y∂x=∂∂y(∂f∂x)=(fx)y=fxy=∂yxf=∂y∂xf.displaystyle frac partial ^2fpartial ypartial x=frac partial partial yleft(frac partial fpartial xright)=(f_x)_y=f_xy=partial _yxf=partial _ypartial _xf.

Higher-order partial and mixed derivatives:

∂i+j+kf∂xi∂yj∂zk=f(i,j,k)=∂xi∂yj∂zkf.displaystyle frac partial ^i+j+kfpartial x^ipartial y^jpartial z^k=f^(i,j,k)=partial _x^ipartial _y^jpartial _z^kf.

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 fdisplaystyle f with respect to xdisplaystyle x, holding ydisplaystyle y and zdisplaystyle z constant, is often expressed as

(∂f∂x)y,z.displaystyle left(frac partial fpartial xright)_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 by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like

∂f(x,y,z)∂xdisplaystyle frac partial f(x,y,z)partial x

is used for the function, while

∂f(u,v,w)∂udisplaystyle 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,v2)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,v2)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,v2)_(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 Didisplaystyle D_i as the partial derivative symbol with respect to the ith variable. For instance, one would write D1f(17,u+v,v2)displaystyle D_1f(17,u+v,v^2) for the example described above, while the expression D1fdisplaystyle D_1f represents the partial derivative function with respect to the 1st variable.^[2]

For higher order partial derivatives, the partial derivative (function) of Difdisplaystyle D_if with respect to the jth variable is denoted Dj(Dif)=Di,jfdisplaystyle D_j(D_if)=D_i,jf. That is, Dj∘Di=Di,jdisplaystyle D_jcirc 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 implies that Di,j=Dj,idisplaystyle D_i,j=D_j,i as long as comparatively mild regularity conditions on f are satisfied.

Antiderivative analogue

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

∂z∂x=2x+y.displaystyle frac partial zpartial x=2x+y.

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

z=∫∂z∂xdx=x2+xy+g(y)displaystyle z=int frac partial zpartial x,dx=x^2+xy+g(y)

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 xdisplaystyle 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 x2+xy+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 2x+ydisplaystyle 2x+y.

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.

Higher order partial derivatives

Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the function f(x,y,...)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):^{[3]:316–318}

∂2f∂x2≡∂∂f/∂x∂x≡∂fx∂x≡fxx.displaystyle frac partial ^2fpartial x^2equiv partial frac partial f/partial xpartial xequiv frac partial f_xpartial xequiv 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

∂2f∂y∂x≡∂∂f/∂x∂y≡∂fx∂y≡fxy.displaystyle frac partial ^2fpartial y,partial xequiv partial frac partial f/partial xpartial yequiv frac partial f_xpartial yequiv f_xy.

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,

∂2f∂x∂y=∂2f∂y∂xdisplaystyle frac partial ^2fpartial x,partial y=frac partial ^2fpartial y,partial x

or equivalently fxy=fyx.displaystyle f_xy=f_yx.

Own and cross partial derivatives appear in the Hessian matrix which is used in the second order conditions in optimization problems.

See also

Notes

References

External links

• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Partial derivative", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• 
  Partial Derivatives at MathWorld