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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a partial derivative of a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 of several variables is its derivative
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
 with respect to one of those variables with the others held constant (as opposed to the total derivative
Total derivative

In the mathematics of differential calculus, the term total derivative has a number of closely related meanings.* The total derivative of a function, f, of several variables, e.g., t,x,y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative....
, in which all variables are allowed to vary). Partial derivatives are useful in vector calculus
Vector calculus

Vector calculus is a branch of mathematics concerned with derivative and integral of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial derivative and multiple integral....
 and differential geometry.

The partial derivative of a function f with respect to the variable x is written as fx, ?xf, or ?f/?x.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a partial derivative of a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 of several variables is its derivative
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
 with respect to one of those variables with the others held constant (as opposed to the total derivative
Total derivative

In the mathematics of differential calculus, the term total derivative has a number of closely related meanings.* The total derivative of a function, f, of several variables, e.g., t,x,y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative....
, in which all variables are allowed to vary). Partial derivatives are useful in vector calculus
Vector calculus

Vector calculus is a branch of mathematics concerned with derivative and integral of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial derivative and multiple integral....
 and differential geometry.

The partial derivative of a function f with respect to the variable x is written as fx, ?xf, or ?f/?x. The partial-derivative symbol ? is a rounded letter, distinguished from the straight d of total-derivative notation. The notation was introduced by Adrien-Marie Legendre
Adrien-Marie Legendre

Adrien-Marie Legendre was a France mathematician. He made important contributions to statistics, number theory, abstract algebra and mathematical analysis....
 and gained general acceptance after its reintroduction by Carl Gustav Jacob Jacobi.

Introduction

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

It is difficult to describe the derivative of such a function, as there are an infinite number of tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 lines to every point on this surface. 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 x-axis, and those that are parallel to the y-axis.

A good way to find these parallel lines is to treat the other variable as a constant. For example, to find the tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 line of the above function at that is parallel to the x-axis, we treat the y variable as constant. The graph and this plane are shown on the right. On the left, we see the way the function looks on the plane . By finding the derivative
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
 of the equation while assuming that y is a constant, we discover that the equation of the tangent line of ƒ is:



So at , by substitution, we know the slope is 3. We write this as:



at the point ,

or as "The partial derivative of z with respect to x at is 3."

Definition

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 x defines a function, denoted fx, which is a function of one real number. That is,

Once a value of x is chosen, say a, then f(x,y) determines a function fa which sends y to a2 + ay + y2:

In this expression, a is a constant, not a variable, so fa is a function of only one real variable, that being y. 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 y direction:

This is the partial derivative of f with respect to y. Here ? is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ? is sometimes pronounced "del" or "partial" instead of "dee".

In general, the partial derivative of a function f(x1,...,xn) in the direction xi at the point (a1,...,an) is defined to be:

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

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,...xn) on a domain in Euclidean space Rn (e.g., on R2 or R3). In this case f has a partial derivative ?f/?xj with respect to each variable xj. At the point a, these partial derivatives define the vector

This vector is called the gradient
Gradient

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
 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 determines a vector field
Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic field or gravity for...
.

Examples

Cone 3d
Consider the volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
 V of a cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
; it depends on the cone's height
Height

Height is the measurement of vertical distance, but has two meanings in common use. It can either indicate how "tall" something is, or how "high up" it is....
 h and its radius
RADIUS

Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
 r according to the formula

The partial derivative of V with respect to r is

It describes 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 is

and represents the rate with which the volume changes if its height is varied and its radius is kept constant.

Now consider by contrast the total derivative
Total derivative

In the mathematics of differential calculus, the term total derivative has a number of closely related meanings.* The total derivative of a function, f, of several variables, e.g., t,x,y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative....
 of V with respect to r and h. They are, respectively

and

We see that the difference between the total and partial derivative is the elimination of indirect dependencies between variables in the latter.

Now suppose that, for some 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:

Equations involving an unknown function's partial derivatives are called partial differential equation
Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a Relation involving an unknown Function of several independent variables and its partial derivatives with respect to those variables....
s and are common in physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, engineering
Engineering

Engineering is the discipline and profession of applying Technology and science knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and process that safely realize a desired objective and meet specified criteria....
, and other science
Science

In its broadest sense, science refers to any systematic knowledge or practice. In its more usual restricted sense, science refers to a system of acquiring knowledge based on scientific method, as well as to the organized body of knowledge gained through such research....
s and applied disciplines.

Notation

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, and it may be necessary to specify explicitly which variables are being held constant. In fields such as statistical mechanics
Statistical mechanics

Statistical mechanics is the application of probability theory, which includes Mathematics tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force....
, the partial derivative of f with respect to x, holding y and z constant, is often expressed as

Formal definition and properties

Like ordinary derivatives, the partial derivative is defined as a limit
Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
. Let U be an open subset
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
 of Rn and f : U ? R a function. We define the partial derivative of f at the point a = (a1, ..., an) ? U with respect to the i-th variable xi as

Even if all partial derivatives ?f/?xi(a) exist at a given point a, the function need not be continuous
Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
 there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f is totally differentiable
Total derivative

In the mathematics of differential calculus, the term total derivative has a number of closely related meanings.* The total derivative of a function, f, of several variables, e.g., t,x,y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative....
 in that neighborhood and the total derivative is continuous. In this case, we say that f is a C1 function. We can use this fact to generalize for vector valued functions (f : U ? Rm) by carefully using a componentwise argument.

The partial derivative can be seen as another function defined on U and can again be partially differentiated. If all mixed second order partial derivatives are continuous at a point (or on a set), we call f a C2 function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem
Symmetry of second derivatives

In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a function...
:

See also


External links

  • at MathWorld
  • on partial derivatives