TheInfoList

OR:

In mathematics, the derivative of a
function of a real variable In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an interv ...
measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...
is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best
linear approximation In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving o ...
of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real f ...
of several variables, the Jacobian matrix reduces to the
gradient vector In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the grad ...
. The process of finding a derivative is called differentiation. The reverse process is called '' antidifferentiation''. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

# Definition

A
function of a real variable In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an interv ...
is ''differentiable'' at a point of its domain, if its domain contains an open interval containing , and the limit :$L=\lim_\frach$ exists. This means that, for every positive real number $\varepsilon$ (even very small), there exists a positive real number $\delta$ such that, for every such that $, h, < \delta$ and $h\ne 0$ then $f\left(a+h\right)$ is defined, and :$\left, L-\frach\<\varepsilon,$ where the vertical bars denote the absolute value (see
(ε, δ)-definition of limit Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
). If the function is differentiable at , that is if the limit exists, then this limit is called the ''derivative'' of at , and denoted $f\text{'}\left(a\right)$ (read as " prime of ") or $\frac(a)$ (read as "the derivative of with respect to at ", " by at ", or " over at "); see , below.

# Continuity and differentiability

If is
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
at , then must also be continuous at . As an example, choose a point and let be the
step function 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 intervals. Informally speaking, a step function is a piecewise constant function having onl ...
that returns the value 1 for all less than , and returns a different value 10 for all greater than or equal to . cannot have a derivative at . If is negative, then is on the low part of the step, so the secant line from to is very steep, and as tends to zero the slope tends to infinity. If is positive, then is on the high part of the step, so the secant line from to has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolute value function given by is continuous at , but it is not differentiable there. If is positive, then the slope of the secant line from 0 to is one, whereas if is negative, then the slope of the secant line from 0 to is negative one. This can be seen graphically as a "kink" or a "cusp" in the graph at . Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by is not differentiable at . In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative. Most functions that occur in practice have derivatives at all points or at almost every point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points. Under mild conditions, for example if the function is a
monotone function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
or a Lipschitz function, this is true. However, in 1872 Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function. In 1931,
Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an origina ...
proved that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions. Informally, this means that hardly any random continuous functions have a derivative at even one point.

# Derivative as a function

Let be a function that has a derivative at every point in its domain. We can then define a function that maps every point to the value of the derivative of at . This function is written and is called the ''derivative function'' or the ''derivative of'' . Sometimes has a derivative at most, but not all, points of its domain. The function whose value at equals whenever is defined and elsewhere is undefined is also called the derivative of . It is still a function, but its domain may be smaller than the domain of . Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions. If we denote this operator by , then is the function . Since is a function, it can be evaluated at a point . By the definition of the derivative function, . For comparison, consider the doubling function given by ; is a real-valued function of a real number, meaning that it takes numbers as inputs and has numbers as outputs: :$\begin 1 &\mapsto 2,\\ 2 &\mapsto 4,\\ 3 &\mapsto 6. \end$ The operator , however, is not defined on individual numbers. It is only defined on functions: :$\begin D\left(x \mapsto 1\right) &= \left(x \mapsto 0\right),\\ D\left(x \mapsto x\right) &= \left(x \mapsto 1\right),\\ D\left\left(x \mapsto x^2\right\right) &= \left(x \mapsto 2\cdot x\right). \end$ Because the output of is a function, the output of can be evaluated at a point. For instance, when is applied to the square function, , outputs the doubling function , which we named . This output function can then be evaluated to get , , and so on.

# Higher derivatives

Let be a differentiable function, and let be its derivative. The derivative of (if it has one) is written and is called the ''
second derivative In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
of ''. Similarly, the derivative of the second derivative, if it exists, is written and is called the '' third derivative of ''. Continuing this process, one can define, if it exists, the th derivative as the derivative of the th derivative. These repeated derivatives are called ''higher-order derivatives''. The th derivative is also called the derivative of order and denoted . If represents the position of an object at time , then the higher-order derivatives of have specific interpretations in physics. The first derivative of is the object's velocity. The second derivative of is the acceleration. The third derivative of is the jerk. And finally, the fourth through sixth derivatives of are
snap, crackle, and pop Snap, Crackle and Pop are the cartoon mascots of Rice Krispies, a brand of breakfast cereal marketed by Kellogg's. History The gnome characters were originally designed by illustrator Vernon Grant in the early 1930s. The names are onomatopoeia ...
; most applicable to astrophysics. A function need not have a derivative (for example, if it is not continuous). Similarly, even if does have a derivative, it may not have a second derivative. For example, let :$f\left(x\right) = \begin +x^2, & \textx\ge 0 \\ -x^2, & \textx \le 0.\end$ Calculation shows that is a differentiable function whose derivative at $x$ is given by :$f\text{'}\left(x\right) = \begin +2x, & \textx\ge 0 \\ -2x, & \textx \le 0.\end$ is twice the absolute value function at $x$, and it does not have a derivative at zero. Similar examples show that a function can have a th derivative for each non-negative integer but not a th derivative. A function that has successive derivatives is called '' times differentiable''. If in addition the th derivative is continuous, then the function is said to be of differentiability class . (This is a stronger condition than having derivatives, as shown by the second example of .) A function that has infinitely many derivatives is called ''infinitely differentiable'' or ''
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
''. On the real line, every polynomial function is infinitely differentiable. By standard differentiation rules, if a polynomial of degree is differentiated times, then it becomes a constant function. All of its subsequent derivatives are identically zero. In particular, they exist, so polynomials are smooth functions. The derivatives of a function at a point provide polynomial approximations to that function near . For example, if is twice differentiable, then :$f\left(x+h\right) \approx f\left(x\right) + f\text{'}\left(x\right)h + \tfrac f\text{'}\text{'}\left(x\right) h^2$ in the sense that :$\lim_\frac = 0.$ If is infinitely differentiable, then this is the beginning of the Taylor series for evaluated at around .

## Inflection point

A point where the second derivative of a function changes sign is called an ''inflection point''. At an inflection point, the second derivative may be zero, as in the case of the inflection point of the function given by $f\left(x\right) = x^3$, or it may fail to exist, as in the case of the inflection point of the function given by $f\left(x\right) = x^\frac$. At an inflection point, a function switches from being a convex function to being a concave function or vice versa.

# Notation (details)

## Leibniz's notation

The symbols $dx$, $dy$, and $\frac$ were introduced by Gottfried Wilhelm Leibniz in 1675. It is still commonly used when the equation is viewed as a functional relationship between dependent and independent variables. Then the first derivative is denoted by : $\frac,\quad\frac, \text\fracf,$ and was once thought of as an infinitesimal quotient. Higher derivatives are expressed using the notation : $\frac, \quad\frac, \text \fracf$ for the ''n''th derivative of $y = f\left(x\right)$. These are abbreviations for multiple applications of the derivative operator. For example, :$\frac = \frac\left\left(\frac\right\right).$ With Leibniz's notation, we can write the derivative of $y$ at the point $x = a$ in two different ways: : $\left.\frac\_ = \frac\left(a\right).$ Leibniz's notation allows one to specify the variable for differentiation (in the denominator), which is relevant in
partial differentiation 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). Par ...
. It also can be used to write the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
as : $\frac = \frac \cdot \frac.$

## Lagrange's notation

Sometimes referred to as ''prime 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 $f$ is denoted $f\text{'}$. Similarly, the second and third derivatives are denoted :$\left(f\text{'}\right)\text{'}=f\text{'}\text{'}$   and   $\left(f\text{'}\text{'}\right)\text{'}=f.$ To denote the number of derivatives beyond this point, some authors use Roman numerals in superscript, whereas others place the number in parentheses: :$f^$   or   $f^.$ The latter notation generalizes to yield the notation $f^$ for the ''n''th 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

Newton's notation In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with ...
for differentiation, also called the dot notation, places a dot over the function name to represent a time derivative. If $y = f\left(t\right)$, then :$\dot$   and   $\ddot$ denote, respectively, the first and second derivatives of $y$. This notation is used exclusively for derivatives with respect to time or arc length. It is typically used in differential equations in physics and differential geometry. The dot notation, however, becomes unmanageable for high-order derivatives (order 4 or more) and cannot deal with multiple independent variables.

## 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 ''n''th derivative is denoted $D^nf$. If 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 :$D_x y$   or   $D_x f\left(x\right)$, although this subscript is often omitted when the variable ''x'' is understood, for instance when this is the only independent variable present in the expression. Euler's notation is useful for stating and solving
linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
s.

# Rules of computation

The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. 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.

## Rules for basic functions

Here are the rules for the derivatives of the most common basic functions, where ''a'' is a real number. * '' Derivatives of powers'': *: $\fracx^a = ax^.$ * ''
Exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
and logarithmic functions'': *: $\frace^x = e^x.$ *: $\fraca^x = a^x\ln\left(a\right),\qquad a > 0$ *: $\frac\ln\left(x\right) = \frac,\qquad x > 0.$ *: $\frac\log_a\left(x\right) = \frac,\qquad x, a > 0$ * '' Trigonometric functions'': *: $\frac\sin\left(x\right) = \cos\left(x\right).$ *: $\frac\cos\left(x\right) = -\sin\left(x\right).$ *: $\frac\tan\left(x\right) = \sec^2\left(x\right) = \frac = 1 + \tan^2\left(x\right).$ * ''
Inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
'': *:

## Rules for combined functions

Here are some of the most basic rules for deducing the derivative of a compound function from derivatives of basic functions. * ''Constant rule'': if ''f''(''x'') is constant, then *: $f\text{'}\left(x\right) = 0.$ * '' Sum rule'': *: $\left(\alpha f + \beta g\right)\text{'} = \alpha f\text{'} + \beta g\text{'}$ for all functions ''f'' and ''g'' and all real numbers ''$\alpha$'' and ''$\beta$''. * ''
Product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
'': *: $\left(fg\right)\text{'} = f \text{'}g + fg\text{'}$ for all functions ''f'' and ''g''. As a special case, this rule includes the fact $\left(\alpha f\right)\text{'} = \alpha f\text{'}$ whenever $\alpha$ is a constant, because $\alpha\text{'} f = 0 \cdot f = 0$ by the constant rule. * ''
Quotient rule In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the deriva ...
'': *: $\left\left(\frac \right\right)\text{'} = \frac$ for all functions ''f'' and ''g'' at all inputs where . * ''
Chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
'' for composite functions: If $f\left(x\right) = h\left(g\left(x\right)\right)$, then *: $f\text{'}\left(x\right) = h\text{'}\left(g\left(x\right)\right) \cdot g\text{'}\left(x\right).$

## Computation example

The derivative of the function given by : $f\left(x\right) = x^4 + \sin \left\left(x^2\right\right) - \ln\left(x\right) e^x + 7$ is : $\begin f\text{'}\left(x\right) &= 4 x^+ \frac\cos \left\left(x^2\right\right) - \frac e^x - \ln\left(x\right) \frac + 0 \\ &= 4x^3 + 2x\cos \left\left(x^2\right\right) - \frac e^x - \ln\left(x\right) e^x. \end$ Here the second term was computed using the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
and third using the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
. The known derivatives of the elementary functions ''x''2, ''x''4, sin(''x''), ln(''x'') and , as well as the constant 7, were also used.

# Definition with hyperreals

Relative to a
hyperreal Hyperreal may refer to: * Hyperreal numbers, an extension of the real numbers in mathematics that are used in non-standard analysis * Hyperreal.org, a rave culture website based in San Francisco, US * Hyperreality, a term used in semiotics and po ...
extension of the real numbers, the derivative of a real function at a real point can be defined as the shadow of the quotient for infinitesimal , where . Here the natural extension of to the hyperreals is still denoted . Here the derivative is said to exist if the shadow is independent of the infinitesimal chosen.

# In higher dimensions

## Vector-valued functions

A vector-valued function y of a real variable sends real numbers to vectors in some vector space R''n''. A vector-valued function can be split up into its coordinate functions , meaning that . This includes, for example,
parametric curve In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
s 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 Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
, called the
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
, whose coordinates are the derivatives of the coordinate functions. That is, :$\mathbf\text{'}\left(t\right) = \left(y\text{'}_1\left(t\right), \ldots, y\text{'}_n\left(t\right)\right).$ Equivalently, :$\mathbf\text{'}\left(t\right)=\lim_\frac,$ if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of y exists for every value of ''t'', then y′ is another vector-valued function. If is the standard basis for R''n'', then y(''t'') can also be written as . If we assume that the derivative of a vector-valued function retains the
linearity Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
property, then the derivative of y(''t'') must be :$y\text{'}_1\left(t\right)\mathbf_1 + \cdots + y\text{'}_n\left(t\right)\mathbf_n$ 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''.

## Partial derivatives

Suppose that ''f'' is a function that depends on more than one variable—for instance, :$f\left(x,y\right) = x^2 + xy + y^2.$ ''f'' can be reinterpreted as a family of functions of one variable indexed by the other variables: :$f\left(x,y\right) = f_x\left(y\right) = x^2 + xy + y^2.$ In other words, every value of ''x'' chooses a function, denoted ''fx'', which is a function of one real number. That is, :$x \mapsto f_x,$ :$f_x\left(y\right) = x^2 + xy + y^2.$ Once a value of ''x'' is chosen, say ''a'', then determines a function ''fa'' that sends ''y'' to : :$f_a\left(y\right) = a^2 + ay + y^2.$ In this expression, ''a'' is a ''constant'', not a ''variable'', so ''fa'' is a function of only one real variable. Consequently, the definition of the derivative for a function of one variable applies: :$f_a\text{'}\left(y\right) = a + 2y.$ The above procedure can be performed for any choice of ''a''. Assembling the derivatives together into a function gives a function that describes the variation of ''f'' in the ''y'' direction: :$\frac\left(x,y\right) = x + 2y.$ This is the partial derivative of ''f'' with respect to ''y''. Here
The Character (symbol), character ∂ (Unicode: U+2202) is a stylized cursive ''d'' mainly used as a Table of mathematical symbols, mathematical symbol, usually to denote a partial derivative such as / (read as "the partial derivative of ''z'' wit ...
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 in the direction ''xi'' at the point (''a''1, ..., ''a''''n'') is defined to be: :$\frac\left(a_1,\ldots,a_n\right) = \lim_\frac.$ In the above difference quotient, all the variables except ''xi'' are held fixed. That choice of fixed values determines a function of one variable :$f_\left(x_i\right) = f\left(a_1,\ldots,a_,x_i,a_,\ldots,a_n\right),$ and, by definition, :$\frac\left(a_i\right) = \frac\left(a_1,\ldots,a_n\right).$ 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. This is fundamental for the study of the functions of several real variables. Let be such a
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real f ...
. If all partial derivatives of are defined at the point , these partial derivatives define the vector :$\nabla f\left(a_1, \ldots, a_n\right) = \left\left(\frac\left(a_1, \ldots, a_n\right), \ldots, \frac\left(a_1, \ldots, a_n\right)\right\right),$ which is called the gradient of at . If is differentiable at every point in some domain, then the gradient is a vector-valued function that maps the point to the vector . Consequently, the gradient determines a vector field.

## Directional derivatives

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 . These are measured using directional derivatives. Choose a vector :$\mathbf = \left(v_1,\ldots,v_n\right).$ The directional derivative of ''f'' in the direction of v at the point x is the limit :$D_\left(\mathbf\right) = \lim_.$ In some cases it may be easier to compute or estimate the directional derivative after changing the length of the vector. Often this is done to turn the problem into the computation of a directional derivative in the direction of a unit vector. To see how this works, suppose that where u is a unit vector in the direction of v. Substitute into the difference quotient. The difference quotient becomes: :$\frac = \lambda\cdot\frac.$ This is ''λ'' times the difference quotient for the directional derivative of ''f'' with respect to u. Furthermore, taking the limit as ''h'' tends to zero is the same as taking the limit as ''k'' tends to zero because ''h'' and ''k'' are multiples of each other. Therefore, . Because of this rescaling property, directional derivatives are frequently considered only for unit vectors. 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: :$D_\left(\boldsymbol\right) = \sum_^n v_j \frac.$ This is a consequence of the definition of the
total derivative In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with res ...
. It follows that the directional derivative is linear in v, meaning that . The same definition also works when ''f'' is a function with values in R''m''. The above definition is applied to each component of the vectors. In this case, the directional derivative is a vector in R''m''.

## Total derivative, total differential and Jacobian matrix

When ''f'' is a function from an open subset of R''n'' to R''m'', then the directional derivative of ''f'' in a chosen direction is the best linear approximation to ''f'' at that point and in that direction. But when , no single directional derivative can give a complete picture of the behavior of ''f''. The total derivative gives a complete picture by considering all directions at once. That is, for any vector v starting at a, the linear approximation formula holds: :$f\left(\mathbf + \mathbf\right) \approx f\left(\mathbf\right) + f\text{'}\left(\mathbf\right)\mathbf.$ Just like the single-variable derivative, is chosen so that the error in this approximation is as small as possible. If ''n'' and ''m'' are both one, then the derivative is a number and the expression is the product of two numbers. But in higher dimensions, it is impossible for to be a number. If it were a number, then would be a vector in R''n'' while the other terms would be vectors in R''m'', and therefore the formula would not make sense. For the linear approximation formula to make sense, must be a function that sends vectors in R''n'' to vectors in R''m'', and must denote this function evaluated at v. To determine what kind of function it is, notice that the linear approximation formula can be rewritten as :$f\left(\mathbf + \mathbf\right) - f\left(\mathbf\right) \approx f\text{'}\left(\mathbf\right)\mathbf.$ Notice that if we choose another vector w, then this approximate equation determines another approximate equation by substituting w for v. It determines a third approximate equation by substituting both w for v and for a. By subtracting these two new equations, we get :$f\left(\mathbf + \mathbf + \mathbf\right) - f\left(\mathbf + \mathbf\right) - f\left(\mathbf + \mathbf\right) + f\left(\mathbf\right) \approx f\text{'}\left(\mathbf + \mathbf\right)\mathbf - f\text{'}\left(\mathbf\right)\mathbf.$ If we assume that v is small and that the derivative varies continuously in a, then is approximately equal to , and therefore the right-hand side is approximately zero. The left-hand side can be rewritten in a different way using the linear approximation formula with substituted for v. The linear approximation formula implies: :$\begin 0 &\approx f\left(\mathbf + \mathbf + \mathbf\right) - f\left(\mathbf + \mathbf\right) - f\left(\mathbf + \mathbf\right) + f\left(\mathbf\right) \\ &= \left(f\left(\mathbf + \mathbf + \mathbf\right) - f\left(\mathbf\right)\right) - \left(f\left(\mathbf + \mathbf\right) - f\left(\mathbf\right)\right) - \left(f\left(\mathbf + \mathbf\right) - f\left(\mathbf\right)\right) \\ &\approx f\text{'}\left(\mathbf\right)\left(\mathbf + \mathbf\right) - f\text{'}\left(\mathbf\right)\mathbf - f\text{'}\left(\mathbf\right)\mathbf. \end$ This suggests that is a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
from the vector space R''n'' to the vector space R''m''. In fact, it is possible to make this a precise derivation by measuring the error in the approximations. Assume that the error in these linear approximation formula is bounded by a constant times , , v, , , where the constant is independent of v but depends continuously on a. Then, after adding an appropriate error term, all of the above approximate equalities can be rephrased as inequalities. In particular, is a linear transformation up to a small error term. In the limit as v and w tend to zero, it must therefore be a linear transformation. Since we define the total derivative by taking a limit as v goes to zero, must be a linear transformation. In one variable, the fact that the derivative is the best linear approximation is expressed by the fact that it is the limit of difference quotients. However, the usual difference quotient does not make sense in higher dimensions because it is not usually possible to divide vectors. In particular, the numerator and denominator of the difference quotient are not even in the same vector space: The numerator lies in the codomain R''m'' while the denominator lies in the domain R''n''. Furthermore, the derivative is a linear transformation, a different type of object from both the numerator and denominator. To make precise the idea that is the best linear approximation, it is necessary to adapt a different formula for the one-variable derivative in which these problems disappear. If , then the usual definition of the derivative may be manipulated to show that the derivative of ''f'' at ''a'' is the unique number such that :$\lim_ \frac = 0.$ This is equivalent to :$\lim_ \frac = 0$ because the limit of a function tends to zero if and only if the limit of the absolute value of the function tends to zero. This last formula can be adapted to the many-variable situation by replacing the absolute values with norms. The definition of the total derivative of ''f'' at a, therefore, is that it is the unique linear transformation such that :$\lim_ \frac = 0.$ Here h is a vector in R''n'', so the norm in the denominator is the standard length on R''n''. However, ''f''′(a)h is a vector in R''m'', and the norm in the numerator is the standard length on R''m''. If ''v'' is a vector starting at ''a'', then is called the
pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
of v by ''f'' and is sometimes written . If the total derivative exists at a, then all the partial derivatives and directional derivatives of ''f'' exist at a, and for all v, is the directional derivative of ''f'' in the direction v. If we write ''f'' using coordinate functions, so that , then the total derivative can be expressed using the partial derivatives as a matrix. This matrix is called the Jacobian matrix of ''f'' at a: :$f\text{'}\left(\mathbf\right) = \operatorname_ = \left\left(\frac\right\right)_.$ The existence of the total derivative ''f''′(a) is strictly stronger than the existence of all the partial derivatives, but if the partial derivatives exist and are continuous, then the total derivative exists, is given by the Jacobian, and depends continuously on a. The definition of the total derivative subsumes the definition of the derivative in one variable. That is, if ''f'' is a real-valued function of a real variable, then 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 is approximately zero, in other words that :$f\left(a+h\right) \approx f\left(a\right) + f\text{'}\left(a\right)h.$ Up to changing variables, this is the statement that the function $x \mapsto f\left(a\right) + f\text{'}\left(a\right)\left(x-a\right)$ is the best linear approximation to ''f'' at ''a''. The total derivative of a function does not give another function in the same way as the 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 In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of the source to the tangent bundle of the target. The natural analog of second, third, and higher-order total derivatives is not a linear transformation, is not a function on the tangent bundle, and is not built by repeatedly taking the total derivative. The analog of a higher-order derivative, called a jet, cannot be a linear transformation because higher-order derivatives reflect subtle geometric information, such as concavity, which cannot be described in terms of linear data such as vectors. It cannot be a function on the tangent bundle because the tangent bundle only has room for the base space and the directional derivatives. Because jets capture higher-order information, they take as arguments additional coordinates representing higher-order changes in direction. The space determined by these additional coordinates is called the jet bundle. The relation between the total derivative and the partial derivatives of a function is paralleled in the relation between the ''k''th order jet of a function and its partial derivatives of order less than or equal to ''k''. By repeatedly taking the total derivative, one obtains higher versions of the
Fréchet derivative In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued ...
, specialized to R''p''. The ''k''th order total derivative may be interpreted as a map :$D^k f: \mathbb^n \to L^k\left(\mathbb^n \times \cdots \times \mathbb^n, \mathbb^m\right)$ which takes a point x in R''n'' and assigns to it an element of the space of ''k''-linear maps from R''n'' to R''m'' – the "best" (in a certain precise sense) ''k''-linear approximation to ''f'' at that point. By precomposing it with the diagonal map Δ, , a generalized Taylor series may be begun as :$\begin f\left(\mathbf\right) & \approx f\left(\mathbf\right) + \left(D f\right)\left(\mathbf\right) + \left\left(D^2 f\right\right)\left(\Delta\left(\mathbf\right)\right) + \cdots\\ & = f\left(\mathbf\right) + \left(D f\right)\left(\mathbf\right) + \left\left(D^2 f\right\right)\left(\mathbf, \mathbf\right)+ \cdots\\ & = f\left(\mathbf\right) + \sum_i \left(D f\right)_i \left(x_i-a_i\right) + \sum_ \left\left(D^2 f\right\right)_ \left(x_j-a_j\right) \left(x_k-a_k\right) + \cdots \end$ where f(a) is identified with a constant function, are the components of the vector , and and are the components of and as linear transformations.

# 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 In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving o ...
of the function at that point. * 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. If C is identified with R2 by writing a complex number ''z'' as , 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 In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differenti ...
– see holomorphic functions. * Another generalization concerns functions between differentiable or smooth manifolds. Intuitively speaking such a manifold ''M'' is a space that can be approximated near each point ''x'' by a vector space called its
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
: the prototypical example is a
smooth surface In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspective ...
in R3. The derivative (or differential) of a (differentiable) map 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 bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
s of ''M'' and ''N''. This definition is fundamental in differential geometry and has many uses – see
pushforward (differential) In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of ''φ, d\varphi_x,'' at a point ''x'' is, in some sense, the be ...
and pullback (differential geometry). * Differentiation can also be defined for maps between infinite dimensional vector spaces such as Banach spaces and
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
s. There is a generalization both of the directional derivative, called the
Gateaux derivative In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died young in World War I, it is defined ...
, and of the differential, called the
Fréchet derivative In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued ...
. * One deficiency of the classical derivative is that very many functions are not 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 In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A n ...
. * The discrete equivalent of differentiation is
finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...
s. The study of differential calculus is unified with the calculus of finite differences in
time scale calculus In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hy ...
. * Also see
arithmetic derivative In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis. T ...
.

# History

Calculus, known in its early history as ''infinitesimal calculus'', is a
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
discipline focused on
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
, functions, derivatives, integrals, and
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
. Isaac Newton and
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
independently discovered calculus in the mid-17th century. However, each inventor claimed the other stole his work in a bitter dispute that continued until the end of their lives.

* Applications of derivatives *
Automatic differentiation In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation, computational differentiation, auto-differentiation, or simply autodiff, is a set of techniques to evaluate the derivative of a function s ...
* Differentiability class * Differentiation rules * Differintegral *
Fractal derivative In applied mathematics and mathematical analysis, the fractal derivative or Hausdorff derivative is a non-Newtonian generalization of the derivative dealing with the measurement of fractals, defined in fractal geometry. Fractal derivatives were ...
*
Generalizations of the derivative In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. Fréchet derivative The Fréchet ...
* Hasse derivative * History of calculus * Integral * Infinitesimal * Linearization *
Mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
* Multiplicative inverse *
Numerical differentiation In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function. Finite differences The simp ...
*
Rate (mathematics) In mathematics, a rate is the ratio between two related quantities in different units. If the denominator of the ratio is expressed as a single unit of one of these quantities, and if it is assumed that this quantity can be changed systematicall ...
*
Radon–Nikodym theorem In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurab ...
*
Symmetric derivative In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined asThomson, p. 1. : \lim_ \frac. The expression under the limit is sometimes called the symmetric difference quotient. A function is said ...
*
Schwarzian derivative In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms an ...

# Bibliography

## Print

* * * * * * * * *

## Online books

* * * * * * * * * *