The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.

In calculus, a branch of mathematics, the derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point. For example, the derivative of the position or distance of a car at some point in time is the instantaneous velocity, or instantaneous speed (respectively), at which that car is traveling (conversely the integral of the velocity is the car's position). For other uses, see Calculus (disambiguation). ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... This article is about functions in mathematics. ... This article is about the concept of integrals in calculus. ...

A closely related notion is the differential of a function. The differential dy In calculus, a differential is an infinitesimally small change in a variable. ...

The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization.^[1] Linear approximation is a method of approximating otherwise difficult to find values of a mathematical function by taking the value on a nearby tangent line instead of the function itself. ... In mathematics, a function of a real variable is a mathematical function whose domain is the real line. ... This article is about the mathematical term. ... For other uses, see tangent (disambiguation). ... relation graph theory In mathematics, the graph of a function f is the collection of all ordered pairs (x,f(x)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... Linearization in mathematics and its applications in general refers to finding the linear approximation to a function at a given point. ...

The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration. The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... In calculus, the integral of a function is an extension of the concept of a sum. ...

Differentiation and the derivative

At each point, the derivative is the slope of a line that is tangent to the curve. The red line is always tangent to the blue curve; its slope is the derivative.

Differentiation is a method to compute the rate at which a quantity, y, changes with respect to the change in another quantity, x, upon which it is dependent. This rate of change is called the derivative of y with respect to x. In more precise language, the dependency of y on x means that y is a function of x. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point. This functional relationship is often denoted y = f(x), where f denotes the function. This article is about the mathematical term. ... Look up line in Wiktionary, the free dictionary. ... For other uses, see tangent (disambiguation). ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In experimental design, a dependent variable (also known as response variable, responding variable or regressand) is a factor whose values in different treatment conditions are compared. ... This article is about functions in mathematics. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... This article is about the mathematical term. ...

The simplest case is when y is a linear function of x, meaning that the graph of y against x is a straight line. In this case, y = f(x) = m x + c, for real numbers m and c, and the slope m is given by A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ...

where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in." This formula is true because Look up Δ, δ in Wiktionary, the free dictionary. ...

y + Δy = f(x+ Δx) = m (x + Δx) + c = m x + c + m Δx = y + mΔx.

It follows that Δy = m Δx.

This gives an exact value for the slope of a straight line. If the function f is not linear (i.e. its graph is not a straight line), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

Figure 1. The tangent line at (x, f(x))

Figure 2. The secant to curve y= f(x) determined by points (x, f(x)) and (x+h, f(x+h)).

Figure 3. The tangent line as limit of secants.

The idea, illustrated by Figures 1-3, is to compute the rate of change as the limiting value of the ratio of the differences Δy / Δx as Δx becomes infinitely small. Download high resolution version (823x586, 6 KB) tangent e= File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (823x586, 6 KB) tangent e= File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... For other uses, see tangent (disambiguation). ... Download high resolution version (823x586, 6 KB) Secant illustration File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (823x586, 6 KB) Secant illustration File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A secant line of a curve is a line that intersects two or more points on the curve. ... Download high resolution version (783x653, 7 KB) Limit of secants File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (783x653, 7 KB) Limit of secants File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In mathematics, the limit of a function is a fundamental concept in analysis. ... The primary vehicle of calculus and other higher mathematics is the function. ...

In Leibniz's notation, such an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written In calculus, Leibnizs notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz, was originally the use of expressions such as dx and dy and to represent infinitely small (or infinitesimal) increments of quantities x and y, just as Δx and Δy represent finite... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ...

suggesting the ratio of two infinitesimal quantities. (The above expression is pronounced in various ways such as "d y by d x" or "d y over d x". The oral form "d y d x" is often used conversationally, although it may lead to confusion.)

The most common approach^[2] to turn this intuitive idea into a precise definition uses limits, but there are other methods, such as non-standard analysis.^[3] Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural...

Definition via difference quotients

Let y=f(x) be a function of x. In classical geometry, the tangent line at a real number a was the unique line through the point (a, f(a)) which did not meet the graph of f transversally, meaning that the line did not pass straight through the graph. The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at a. The slope of the tangent line is very close to the slope of the line through (a, f(a)) and a nearby point on the graph, for example (a + h, f(a + h)). These lines are called secant lines. A value of h close to zero will give a good approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations. The slope of the secant line is the difference between the y values of these points divided by the difference between the x values, that is, Transversality in mathematics is a notion that describes how spaces can intersect; transversality can be seen as the opposite of tangency, and plays a role in general position. ... A secant line of a curve is a line that intersects two or more points on the curve. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... It has been suggested that this article or section be merged with estimation. ...

This expression is Newton's difference quotient. The derivative is the value of the difference quotient as the secant lines get closer and closer to the tangent line. Formally, the derivative of the function f at a is the limit Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... The primary vehicle of calculus and other higher mathematics is the function. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as...

of the difference quotient as h approaches zero, if this limit exists. If the limit exists, then f is differentiable at a. Here f′ (a) is one of several common notations for the derivative (see below). For other uses, see Derivative (disambiguation). ...

Equivalently, the derivative satisfies the property that

which has the intuitive interpretation (see Figure 1) that the tangent line to f at a gives the best linear approximation For other uses, see Linear (disambiguation). ...

to f near a (i.e., for small h). This interpretation is the easiest to generalize to other settings (see below). For other uses, see Derivative (disambiguation). ...

Substituting 0 for h in the difference quotient causes division by zero, so the slope of the tangent line cannot be found directly. Instead, define Q(h) to be the difference quotient as a function of h: See also the disambiguation page title equality. ... For the album by Hux Flux, see Division by Zero (album). ...

.

Q(h) is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from the point h = 0. If the limit exists, meaning that there is a way of choosing a value for Q(0) which makes the graph of Q a continuous function, then the function f is differentiable at the point a, and its derivative at a equals Q(0). In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

In practice, the continuity of the difference quotient Q(h) at h = 0 is shown by modifying the numerator to cancel h in the denominator. This process can be long and tedious for complicated functions, and many short cuts are commonly used to simplify the process. For other uses, see Derivative (disambiguation). ...

Example

The squaring function f(x) = x² is differentiable at x = 3, and its derivative there is 6. This is proven by writing the difference quotient as follows:

Then we get the simplified function in the limit:

The last expression shows that the difference quotient equals 6 + h when h is not zero and is undefined when h is zero. (Remember that because of the definition of the difference quotient, the difference quotient is always undefined when h is zero.) However, there is a natural way of filling in a value for the difference quotient at zero, namely 6. Hence the slope of the graph of the squaring function at the point (3, 9) is 6, and so its derivative at x = 3 is f '(3) = 6.

More generally, a similar computation shows that the derivative of the squaring function at x = a is f '(a) = 2a.

Continuity and differentiability

This function does not have a derivative at the marked point, as the function is not continuous there.

If y = f(x) is differentiable at a, then f must also be continuous at a. As an example, choose a point a and let f be the step function which returns a value, say 1, for all x less than a, and returns a different value, say 10, for all x greater than or equal to a. f cannot have a derivative at a. If h is negative, then a + h is on the low part of the step, so the secant line from a to a + h will be very steep, and as h tends to zero the slope tends to infinity. If h is positive, then a + h is on the high part of the step, so the secant line from a to a + h will have slope zero. Consequently the secant lines do not approach any single slope, so the limit of the difference quotient does not exist.^[4] Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of half-open intervals. ...

The absolute value function is continuous, but fails to be differentiable at x = 0 since it has a sharp corner.

However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolute value function y = |x| is continuous at x = 0, but it is not differentiable there. If h is positive, then the slope of the secant line from 0 to h is one, whereas if h is negative, then the slope of the secant line from 0 to h is negative one. This can be seen graphically as a "kink" in the graph at x = 0. Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function y = ³√x is not differentiable at x = 0. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...

Most functions which occur in practice have derivatives at all points or at almost every point. However, a result of Stefan Banach states that the set of functions which have a derivative at some point is a meager set in the space of all continuous functions.^[5] Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function. In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... Stefan Banach ( listen) (Ukrainian: Степан Степанович Банах, 1892-1945) was an eminent Polish mathematician who worked in interwar Poland and briefly in Soviet Ukraine. ... In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ... In BIOLOGY, the SUMMER VACATION function was the first example found of a Chumba wumbafunction with the property that it is continuous everywhere but differentiable nowhere. ...

The derivative as a function

Let f be a function that has a derivative at every point a in the domain of f. Because every point a has a derivative, there is a function which sends the point a to the derivative of f at a. This function is written f′(x) and is called the derivative function or the derivative of f. The derivative of f collects all the derivatives of f at all the points in the domain of f. In mathematics, the domain of a function is the set of all input values to the function. ...

Sometimes f has a derivative at most, but not all, points of its domain. The function whose value at a equals f′(a) whenever f′(a) is defined and is undefined elsewhere is also called the derivative of f. It is still a function, but its domain is strictly smaller than the domain of f.

Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions which have derivatives at every point of their domain and whose range is a set of functions. If we denote this operator by D, then D(f) is the function f′(x). Since D(f) is a function, it can be evaluated at a point a. By the definition of the derivative function, D(f)(a) = f′(a). In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...

For comparison, consider the doubling function f(x) =2x; f is a real-valued function of a real number, meaning that it takes numbers as inputs and has numbers as outputs:

The operator D, however, is not defined on individual numbers. It is only defined on functions:

Because the output of D is a function, the output of D can be evaluated at a point. For instance, when D is applied to the squaring function,

D outputs the doubling function,

which we named f(x). This output function can then be evaluated to get f(1) = 2, f(2) = 4, and so on.

Higher derivatives

Let f be a differentiable function, and let f′(x) be its derivative. The derivative of f′(x) (if it has one) is written f′′(x) and is called the second derivative of f. Similarly, the derivative of a second derivative, if it exists, is written f′′′(x) and is called the third derivative of f. These repeated derivatives are called higher-order derivatives.

A function f need not have a derivative, for example, if it is not continuous. Similarly, even if f does have a derivative, it may not have a second derivative. For example, let

.

An elementary calculation shows that f is a differentiable function whose derivative is

.

f′(x) is twice the absolute value function, and it does not have a derivative at zero. Similar examples show that a function can have k derivatives for any non-negative integer k but no (k + 1)-order derivative. A function that has k successive derivatives is called k times differentiable. If in addition the kth derivative is continuous, then the function is said to be of differentiability class C^k. (This is a stronger condition than having k derivatives. For an example, see differentiability class.) A function that has infinitely many derivatives is called infinitely differentiable or smooth. A differentiability class in mathematics is a class of functions which share differentiability features. ... A differentiability class in mathematics is a class of functions which share differentiability features. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ...

On the real line, every polynomial function is infinitely differentiable. By standard differentiation rules, if a polynomial of degree n is differentiated n times, then it becomes a constant function. All of its subsequent derivatives are identically zero. In particular, they exist, so polynomials are smooth functions. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. ... In mathematics a constant function is a function whose values do not vary and thus are constant. ...

The derivatives of a function f at a point x provide polynomial approximations to that function near x. For example, if f is twice differentiable, then

in the sense that

If f is infinitely differentiable, then this is the beginning of the Taylor series for f. Series expansion redirects here. ...

Notations for differentiation

Main article: Notation for differentiation

This article or section does not adequately cite its references or sources. ...

Leibniz's notation

Main article: Leibniz's notation

The notation for derivatives introduced by Gottfried Leibniz is one of the earliest. It is still commonly used when the equation y=f(x) is viewed as a functional relationship between dependent and independent variables. Then the first derivative is denoted by In calculus, Leibnizs notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz, was originally the use of expressions such as dx and dy and to represent infinitely small (or infinitesimal) increments of quantities x and y, just as Δx and Δy represent finite... Leibniz redirects here. ... Dependent and independent variables refer to values that change in relationship to each other. ...

Higher derivatives are expressed using the notation

for the nth derivative of y = f(x) (with respect to x).

With Leibniz's notation, we can write the derivative of y at the point x = a in two different ways:

Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember:^[6] 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). ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

Lagrange's notation

One of the most common modern notations for differentiation is due to Joseph Louis Lagrange and uses the prime mark, so that the derivative of a function f(x) is denoted f′(x) or simply f′. Similarly, the second and third derivatives are denoted Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. ... This article is not about the symbol for the set of prime numbers, â„™. The prime (′, Unicode U+2032, ′) is a symbol with many mathematical uses: A complement in set theory: A′ is the complement of the set A A point related to another (e. ...

and

Beyond this point, some authors use Roman numerals such as

for the fourth derivative, whereas other authors place the number of derivatives in parentheses:

The latter notation generalizes to yield the notation f ⁽ⁿ⁾ for the nth derivative of f — this notation is most useful when we wish to talk about the derivative as being a function itself, as in this case the Leibniz notation can become cumbersome.

Newton's notation

Main article: Newton's notation

Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a derivative. If y = f(t), then Newtons notation for differentiation involved placing a dash/dot over the function name, which he termed the fluxion. ... Newtons notation for differentiation involved placing a dash/dot over the function name, which he termed the fluxion. ...

and

denote, respectively, the first and second derivatives of y with respect to t. This notation is used almost exclusively for time derivatives, meaning that the independent variable of the function represents time. It is very common in physics and in mathematical disciplines connected with physics such as differential equations. While the notation becomes unmanageable for high-order derivatives, in practice only very few derivatives are needed. A time derivative is a derivative of a function with respect to time, t. ... This article is about the concept of time. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ...

Euler's notation

Euler's notation uses a differential operator D, which is applied to a function f to give the first derivative Df. The second derivative is denoted D²f, and the nth derivative is denoted Dⁿf. Euler redirects here. ... In mathematics, a differential operator is an operator defined as a function of the differentiation operator. ...

If y = f(x) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x. Euler's notation is then written

or ,

although this subscript is often omitted when the variable x is understood, for instance when this is the only variable present in the expression.

Euler's notation is useful for stating and solving linear differential equations. In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ...

Computing the derivative

The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. For some examples, see Derivative (examples). In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones. // Consider f(x) = 5: The derivative of a constant function is zero. ...

Derivatives of elementary functions

Main article: Table of derivatives

In addition, the derivatives of some common functions are useful to know. The primary operation in differential calculus is finding a derivative. ...

• Derivatives of powers: if

,

where r is any real number, then In mathematics, polynomials are perhaps the simplest functions with which to do calculus. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

,

wherever this function is defined. For example, if r = 1/2, then

.

and the function is defined only for non-negative x. When r = 0, this rule recovers the constant rule.

• Exponential and logarithm functions:

• Trigonometric functions:

• Inverse trigonometric functions:

The exponential function is one of the most important functions in mathematics. ... Look up logarithm in Wiktionary, the free dictionary. ... Sine redirects here. ... In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ...

Rules for finding the derivative

Main article: Differentiation rules

In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided using differentiation rules. Some of the most basic rules are the following. This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. ...

• Constant rule: if f(x) is constant, then

• Sum rule:

for all functions f and g and all real numbers a and b.

• Product rule:

for all functions f and g.

• Quotient rule:

• Chain rule: If f(x) = h(g(x)), then

.

In mathematics, the linearity of differentiation is a most fundamental property of the derivative, in differential calculus. ... In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

Example computation

The derivative of

is

Here the second term was computed using the chain rule and third using the product rule: the known derivatives of the elementary functions x², x⁴, sin(x), ln(x) and exp(x) = e^x were also used.

Derivatives in higher dimensions

See also: vector calculus and multivariable calculus

Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... Multivariable calculus is the extension of calculus in one variable to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable. ...

Derivatives of vector valued functions

A vector-valued function y(t) of a real variable is a function which sends real numbers to vectors in some vector space Rⁿ. A vector-valued function can be split up into its coordinate functions y₁(t), y₂(t), …, y_n(t), meaning that y(t) = (y₁(t), ..., y_n(t)). This includes, for example, parametric curves in R² or R³. The coordinate functions are real valued functions, so the above definition of derivative applies to them. The derivative of y(t) is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is, A graph of the vector-valued function <2Cos(t),4Sin(t),t> A vector-valued function is a mathematical function that maps real numbers onto vectors. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... This article is about vectors that have a particular relation to the spatial coordinates. ... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...

Equivalently,

if the limit exists. The subtraction in the numerator is subtraction of vectors, not scalars. If the derivative of y exists for every value of t, then y′ is another vector valued function.

If e₁, …, e_n is the standard basis for Rⁿ, then y(t) can also be written as y₁(t)e₁ + … + y_n(t)e_n. If we assume that the derivative of a vector-valued function retains the linearity property, then the derivative of y(t) must be In mathematics, the linearity of differentiation is a most fundamental property of the derivative, in differential calculus. ...

because each of the basis vectors is a constant.

This generalization is useful, for example, if y(t) is the position vector of a particle at time t; then the derivative y′(t) is the velocity vector of the particle at time t. This article is about velocity in physics. ...

Partial derivatives

Main article: Partial derivative

Suppose that f is a function that depends on more than one variable. For instance, 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). ...

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

In other words, every value of x chooses a function, denoted f_x, which is a function of one real number.^[7] That is,

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

In this expression, a is a constant, not a variable, so f_a is a function of only one real variable. 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 "der", "del", or "partial" instead of "dee".

In general, the partial derivative of a function f(x₁, …, x_n) in the direction x_i at the point (a₁ …, a_n) is defined to be:

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

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,...x_n) on a domain in Euclidean space Rⁿ (e.g., on R² or R³). 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 In mathematics and physics, a scalar field associates a scalar value, which can be either mathematical in definition, or physical, to every point in space. ...

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 determines a vector field. For other uses, see Gradient (disambiguation). ... Vector field given by vectors of the form (−y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space. ...

Directional derivatives

Main article: Directional derivative

If f is a real-valued function on Rⁿ, then the partial derivatives of f measure its variation in the direction of the coordinate axes. For example, if f is a function of x and y, then its partial derivatives measure the variation in f in the x direction and the y direction. They do not, however, directly measure the variation of f in any other direction, such as along the diagonal line y = x. These are measured using directional derivatives. Choose a vector In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the...

The directional derivative of f in the direction of v at the point x is the limit

Let λ be a scalar. The substitution of h/λ for h changes the λv direction's difference quotient into λ times the v direction's difference quotient. Consequently, the directional derivative in the λv direction is λ times the directional derivative in the v direction. Because of this, directional derivatives are often considered only for unit vectors v.

If all the partial derivatives of f exist and are continuous at x, then they determine the directional derivative of f in the direction v by the formula:

This is a consequence of the definition of the total derivative. It follows that the directional derivative is linear in v. In mathematics, a total derivative may be either. ... In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

The same definition also works when f is a function with values in R^m. We just use the above definition in each component of the vectors. In this case, the directional derivative is a vector in R^m.

The total derivative, the total differential and the Jacobian

Main article: Total derivative

Let f be a function from a domain in R to R. The derivative of f at a point a in its domain is the best linear approximation to f at that point. As above, this is a number. Geometrically, if v is a unit vector starting at a, then f′ (a) , the best linear approximation to f at a, should be the length of the vector found by moving v to the target space using f. (This vector is called the pushforward of v by f and is usually written f _* v.) In other words, if v is measured in terms of distances on the target, then, because v can only be measured through f, v no longer appears to be a unit vector because f does not preserve unit vectors. Instead v appears to have length f′ (a). If m is greater than one, then by writing f using coordinate functions, the length of v in each of the coordinate directions can be measured separately. In mathematics, a total derivative may be either. ... Suppose that φ : M → N is a smooth map between smooth manifolds; then the differential of φ at a point x is, in some sense, the best linear approximation of φ near x. ...

Suppose now that f is a function from a domain in Rⁿ to R^m and that a is a point in the domain of f. The derivative of f at a should still be the best linear approximation to f at a. In other words, if v is a vector on Rⁿ, then f′ (a) should be the linear transformation that best approximates f. The linear transformation should contain all the information about how f transforms vectors at a to vectors at f(a), and in symbols, this means it should be the linear transformation f′ (a) such that In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

Here h is a vector in Rⁿ, so the norm in the denominator is the standard length on Rⁿ. However, f′ (a)h is a vector in R^m, and the norm in the numerator is the standard length on R^m. The linear transformation f′ (a), if it exists, is called the total derivative of f at a or the (total) differential of f at a. In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... The differential dy In calculus, a differential is an infinitesimally small change in a variable. ...

If the total derivative exists at a, then all the partial derivatives of f exist at a. If we write f using coordinate functions, so that f = (f₁, f₂, ..., f_m), then the total derivative can be expressed as a matrix called the Jacobian matrix of f at a: In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... For the French Revolution faction, see Jacobin. ...

The existence of the Jacobian is strictly stronger than existence of all the partial derivatives, but if the partial derivatives exist and satisfy mild smoothness conditions, then the total derivative exists and is given by the Jacobian.

The definition of the total derivative subsumes the definition of the derivative in one variable. In this case, the total derivative exists if and only if the usual derivative exists. The Jacobian matrix reduces to a 1×1 matrix whose only entry is the derivative f′ (x). This 1×1 matrix satisfies the property that f(a + h) − f(a) − f′(a)h is approximately zero, in other words that

Up to changing variables, this is the statement that the function is the best linear approximation to f at a.

The total derivative of a function does not give another function in the same way that one-variable case. This is because the total derivative of a multivariable function has to record much more information than the derivative of a single-variable function. Instead, the total derivative gives a function from the tangent bundle of the source to the tangent bundle of the target. In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...

Generalizations

Main article: Derivative (generalizations)

The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a linear approximation of the function at that point. In mathematics, there are many possible generalizations of the derivative, i. ... Linear approximation is a method of approximating otherwise difficult to find values of a mathematical function by taking the value on a nearby tangent line instead of the function itself. ...

• An important generalization of the derivative concerns complex functions of complex variables, such as functions from (a domain in) the complex numbers C to C. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition. However, this innocent definition hides some very deep properties. If C is identified with R² by writing a complex number z as x + i y, then a differentiable function from C to C is certainly differentiable as a function from R² to R² (in the sense that its partial derivatives all exist), but the converse is not true in general: the complex derivative only exists if the real derivative is complex linear and this imposes relations between the partial derivatives called the Cauchy Riemann equations — see holomorphic functions.

• Another generalization concerns functions between differentiable or smooth manifolds. Intuitively speaking such a manifold M is a space which can be approximated near each point x by a vector space called its tangent space: the prototypical example is a smooth surface in R³. The derivative (or differential) of a (differentiable) map f: M → N between manifolds, at a point x in M, is then a linear map from the tangent space of M at x to the tangent space of N at f(x). The derivative function becomes a map between the tangent bundles of M and N. This definition is fundamental in differential geometry and has many uses — see pushforward (differential) and pullback (differential geometry).

• Differentiation can also be defined for maps between infinite dimensional vector spaces such as Banach spaces and Fréchet spaces. There is a generalization both of the directional derivative, called the Gâteaux derivative, and of the differential, called the Fréchet derivative.

• One deficiency of the classical derivative is that not very many functions are differentiable. Nevertheless, there is a way of extending the notion of the derivative so that all continuous functions and many other functions can be differentiated using a concept known as the weak derivative. The idea is to embed the continuous functions in a larger space called the space of distributions and only require that a function is differentiable "on average".

• The properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology — see, for example, differential algebra.

• Also see arithmetic derivative.

In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, the Cauchy-Riemann differential equations in complex analysis are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ... In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Suppose that φ : M → N is a smooth map between smooth manifolds; then the differential of φ at a point x is, in some sense, the best linear approximation of φ near x. ... Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is... In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... This article deals with Fréchet spaces in functional analysis. ... In mathematics, the Gâteaux derivative is a generalisation in functional analysis of the standard derivative of the differential calculus. ... In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... In mathematics, differential rings, differential fields and differential algebras are rings, fields and algebras equipped with a derivation, which is a unary function satisfying the Leibniz product law. ...

Notes

The differential dy In calculus, a differential is an infinitesimally small change in a variable. ... In mathematics, the Radon-Nikodym theorem is a result in functional analysis that states that if a measure Q is absolutely continuous with respect to another sigma-finite measure P then there is a measurable function f, taking values in [0,∞], on the underlying space such that for any measurable... In mathematics, the Kähler differentials are a universal construction Ω1S/R associated to a ring homomorphism of commutative rings, φ:R → S, that provides an analogue of the construction of differential forms (1-forms). ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... In mathematics, a Dirac measure is a measure δx on a set X that gives a given element x measure 1, so that δx({x}) = 1 and in general δx(Y) = 0 for any subset Y of X not containing x, δx(Z) = 1 for any subset Z containing x. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... The differential dy In calculus, a differential is an infinitesimally small change in a variable. ... In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ...

References

Print

• Anton, Howard; Bivens, Irl & Davis, Stephen (February 2, 2005), Calculus: Early Transcendentals Single and Multivariable (8th ed.), New York: Wiley, ISBN 978-0471472445 
• Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, vol. 1 (2nd ed.), Wiley, ISBN 978-0471000051 
• Apostol, Tom M. (June 1969), Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications, vol. 1 (2nd ed.), Wiley, ISBN 978-0471000075 
• Eves, Howard (January 2, 1990), An Introduction to the History of Mathematics (6th ed.), Brooks Cole, ISBN 978-0030295584 
• Larson, Ron; Hostetler, Robert P. & Edwards, Bruce H. (February 28, 2006), Calculus: Early Transcendental Functions (4th ed.), Houghton Mifflin Company, ISBN 978-0618606245 
• Spivak, Michael (September 1994), Calculus (3rd ed.), Publish or Perish, ISBN 978-0914098898 
• Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0534393397 
• Thompson, Silvanus P. (September 8, 1998), Calculus Made Easy (Revised, Updated, Expanded ed.), New York: St. Martin's Press, ISBN 978-0312185480 

Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. ...

Online books

• Crowell, Benjamin (2003), Calculus, <http://www.lightandmatter.com/calc/> 
• Garrett, Paul (2004), Notes on First-Year Calculus, <http://www.math.umn.edu/~garrett/calculus/> 
• Hussain, Faraz (2006), Understanding Calculus, <http://www.understandingcalculus.com/> 
• Keisler, H. Jerome (2000), Elementary Calculus: An Approach Using Infinitesimals, <http://www.math.wisc.edu/~keisler/calc.html> 
• Mauch, Sean (2004), Unabridged Version of Sean's Applied Math Book, <http://www.its.caltech.edu/~sean/book/unabridged.html> 
• Sloughter, Dan (2000), Difference Equations to Differential Equations, <http://math.furman.edu/~dcs/book/> 
• Strang, Gilbert (1991), Calculus, <http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm> 
• Stroyan, Keith D. (1997), A Brief Introduction to Infinitesimal Calculus, <http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm> 
• Wikibooks, Calculus, <http://en.wikibooks.org/wiki/Calculus> 

See also

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External links

• WIMS Function Calculator makes online calculation of derivatives; this software also enables interactive exercises.
• Online Derivatives Calculator.
• Mathematical Assistant on Web online calculation of derivatives, including explanation of steps in the solution.
• Proving Derivatives from First Principles.
• Practice finding derivatives of randomly generated functions