TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the derivative of a
function of a real variable In mathematical analysis, and applications in geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. ...
measures the sensitivity to change of the function value (output value) with respect to a change in its
argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
(input value). Derivatives are a fundamental tool of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

. For example, the derivative of the position of a moving object with respect to
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

is the object's
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

: 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 In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

of the
tangent line In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

to the at that point. The tangent line is the best
linear approximation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 mathematical analysis its applications, a function of several real variables or real multivariate function is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A comp ...
. In this generalization, the derivative is reinterpreted as a
linear transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix (mathematics), matrix of all its first-order partial derivatives. When this matrix is square matrix, square, that is, when the function t ...
is the
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
that represents this
linear transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s with respect to the independent variables. For a
real-valued function Mass measured in grams is a function from this collection of weight to positive number">positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Converging lens or positive lens, i ...
of several variables, the Jacobian matrix reduces to the
gradient vector In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...
. The process of finding a derivative is called differentiation. The reverse process is called ''
antidifferentiation In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function (mathematics), function is a differentiable function whose derivative is equal to the original function . This can ...
''. The
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating the gradient) with the concept of integral, integrating a function (calculating the area under t ...
relates antidifferentiation with . 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 Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. ...
is ''differentiable'' at a point of its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
, if its domain contains an
open interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
containing , and the
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
:$L=\lim_\frach$ exists. This means that, for every positive
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
$\varepsilon$ (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 In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

(see (ε, δ)-definition of limit). 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.

# Explanations

''Differentiation'' is the action of computing a derivative. The derivative of a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
of a variable is a measure of the rate at which the value of the function changes with respect to the change of the variable . It is called the ''derivative'' of with respect to . If and are
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, and if the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

of is plotted against , derivative is the
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

of this graph at each point. The simplest case, apart from the trivial case of a
constant function 270px, Constant function ''y''=4 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (m ...

, is when is a
linear function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

of , meaning that the graph of is a line. In this case, , for real numbers and , and the slope is given by :$m=\frac = \frac,$ where the symbol (
Delta Delta commonly refers to: * Delta (letter) (Δ or δ), a letter of the Greek alphabet * River delta, a landform at the mouth of a river * D (NATO phonetic alphabet: "Delta"), the fourth letter of the modern English alphabet * Delta Air Lines, an Ame ...
) is an abbreviation for "change in", and the combinations $\Delta x$ and $\Delta y$ refer to corresponding changes, i.e. :$\Delta y = f\left(x + \Delta x\right)- f\left(x\right)$. The above formula holds because :$\begin y + \Delta y &= f\left\left( x+\Delta x\right\right)\\ &= m\left\left( x+\Delta x\right\right) +b =mx +m\Delta x +b \\ &= y + m\Delta x. \end$ Thus :$\Delta y=m\Delta x.$ This gives the value for the slope of a line. If the function is not linear (i.e. its graph is not a straight line), then the change in divided by the change in varies over the considered range: differentiation is a method to find a unique value for this rate of change, not across a certain range $\left(\Delta x\right),$ but at any given value of . The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the of the ratio of the differences as tends towards 0.

## Toward a definition

The most common approach to turn this intuitive idea into a precise definition is to define the derivative as a
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
of difference quotients of real numbers. This is the approach described below. Let be a real valued function defined in an open neighborhood of a real number . In classical geometry, the tangent line to the graph of the function at was the unique line through the point that did ''not'' meet the graph of transversally, meaning that the line did not pass straight through the graph. The derivative of with respect to at is, geometrically, the slope of the tangent line to the graph of at . The slope of the tangent line is very close to the slope of the line through and a nearby point on the graph, for example . These lines are called
secant line In geometry, a secant is a line (geometry), line that intersects a curve at a minimum of two distinct Point (geometry), points.. The word ''secant'' comes from the Latin word ''secare'', meaning ''to cut''. In the case of a circle, a secant inters ...
s. A value of close to zero gives a good approximation to the slope of the tangent line, and smaller values (in
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

) of will, in general, give better
approximation An approximation is anything that is intentionally similar but not exactly equal Equal or equals may refer to: Arts and entertainment * Equals (film), ''Equals'' (film), a 2015 American science fiction film * Equals (game), ''Equals'' (game), a ...
s. The slope of the secant line is the difference between the values of these points divided by the difference between the values, that is, :$m = \frac = \frac = \frac.$ This expression is
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...

's
difference quotient In single-variable calculus, the difference quotient is usually the name for the expression : \frac which when taken to the Limit of a function, limit as ''h'' approaches 0 gives the derivative of the Function (mathematics), function ''f''. The ...
. Passing from an approximation to an exact answer is done using a
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

. Geometrically, the limit of the secant lines is the tangent line. Therefore, the limit of the difference quotient as approaches zero, if it exists, should represent the slope of the tangent line to . This limit is defined to be the derivative of the function at : :$f\text{'}\left(a\right)=\lim_\frac.$ When the limit exists, is said to be ''
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...
'' at . Here is one of several common notations for the derivative (
see below See or SEE may refer to: Arts, entertainment, and media * Music: ** See (album), ''See'' (album), studio album by rock band The Rascals *** "See", song by The Rascals, on the album ''See'' ** See (Tycho song), "See" (Tycho song), song by Tycho * T ...
). From this definition it is obvious that a differentiable function is
increasing Image:Monotonicity example3.png, Figure 3. A function that is not monotonic In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that pres ...

if and only if its derivative is positive, and is decreasing
iff IFF, Iff or iff may refer to: Arts and entertainment * Simon Iff, a fictional character by Aleister Crowley * Iff of the Unpronounceable Name, a fictional character in the Riddle-Master trilogy by Patricia A. McKillip * "IFF", an List of The ...
its derivative is negative. This fact is used extensively when analyzing function behavior, e.g. when finding
local extrema In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function (mathematics), function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, ei ...
. Equivalently, the derivative satisfies the property that :$\lim_\frac = 0,$ which has the intuitive interpretation (see Figure 1) that the tangent line to at gives the ''best
linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...

approximation'' :$f\left(a+h\right) \approx f\left(a\right) + f\text{'}\left(a\right)h$ to near (i.e., for small ). This interpretation is the easiest to generalize to other settings (
see below See or SEE may refer to: Arts, entertainment, and media * Music: ** See (album), ''See'' (album), studio album by rock band The Rascals *** "See", song by The Rascals, on the album ''See'' ** See (Tycho song), "See" (Tycho song), song by Tycho * T ...
). Substituting 0 for in the difference quotient causes
division by zero In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, so the slope of the tangent line cannot be found directly using this method. Instead, define to be the difference quotient as a function of : :$Q\left(h\right) = \frac.$ is the slope of the secant line between and . If is a
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, meaning that its graph is an unbroken curve with no gaps, then is a continuous function away from . If the limit exists, meaning that there is a way of choosing a value for that makes a continuous function, then the function is differentiable at , and its derivative at equals . In practice, the existence of a continuous extension of the difference quotient to is shown by modifying the numerator to cancel in the denominator. Such manipulations can make the limit value of for small clear even though is still not defined at . This process can be long and tedious for complicated functions, and many shortcuts are commonly used to simplify the process.

## Example

The square function given by is differentiable at , and its derivative there is 6. This result is established by calculating the limit as approaches zero of the difference quotient of : :$\begin f\text{'}\left(3\right) & = \lim_\frac = \lim_\frac \\$
0pt PT, Pt, or pt may refer to: Arts and entertainment * , acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * , a British progressive rock group In business Businesses * (IATA ...
& = \lim_\frac = \lim_\frac = \lim_. \end The last expression shows that the difference quotient equals when and is undefined when , because of the definition of the difference quotient. However, the definition of the limit says the difference quotient does not need to be defined when . The limit is the result of letting go to zero, meaning it is the value that tends to as becomes very small: :$\lim_ = 6 + 0 = 6.$ Hence the slope of the graph of the square function at the point is , and so its derivative at is . More generally, a similar computation shows that the derivative of the square function at is : :

# Continuity and differentiability

If is
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...
at , then must also be
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
at . As an example, choose a point and let be the
step function In mathematics, a function on the real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R\$) * Central American Republic real * Mexican ...

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 In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

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 : 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 In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to t ...
point. Early in the
history of calculus Calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They ...
, many mathematicians assumed that a continuous function was differentiable at most points. Under mild conditions, for example if the function is a
monotone function Figure 3. A function that is not monotonic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
or a
Lipschitz function In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function In mathematics Mathematics (from Ancient Greek, Greek: ) i ...
, 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 300px, Plot of Weierstrass function over the interval minus;2, 2 Like other fractals, the function exhibits self-similarity">fractal.html" ;"title="minus;2, 2 Like other fractal">minus;2, 2 Like other fractals, the function exhibi ...

. In 1931,
Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quan ...

proved that the set of functions that have a derivative at some point is a
meager set In the fields of and , a meagre set (also called a meager set or a set of first category) is a that, considered as a of a (usually larger) , is in a precise sense small or . A topological space is called meagre if it is a meager subset of its ...
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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
. 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 is strictly 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 '' 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 . If represents the position of an object at time , then the higher-order derivatives of have specific interpretations in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

. The first derivative of is the object's
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

. The second derivative of is the
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

. The third derivative of is the jerk. And finally, the fourth through sixth derivatives of are snap, crackle, and pop; most applicable to
astrophysics Astrophysics is a science that employs the methods and principles of physics in the study of astronomical objects and phenomena. Among the subjects studied are the Sun, other stars, galaxy, galaxies, extrasolar planets, the interstellar medium and ...
. 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 In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...
. (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 is a smooth function with compact support. In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuo ...
''. On the real line, every
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
is infinitely differentiable. By standard
differentiation rules This is a summary of differentiation rules, that is, rules for computing the of a in . Elementary rules of differentiation Unless otherwise stated, all functions are functions of that return real values; although more generally, the formul ...
, if a polynomial of degree is differentiated times, then it becomes a
constant function 270px, Constant function ''y''=4 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (m ...

. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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 (in green) is a convex set File:Convex polygon illustration2.svg, Illustration of a non-convex set. Illustrated by the above line segment whereby it changes from the black color to the red color. Exemplifying why this above set, illustrated in gr ...

to being a
concave function In , a concave function is the of a . A concave function is also ously called concave downwards, concave down, convex upwards, convex cap, or upper convex. Definition A real-valued f on an (or, more generally, a in ) is said to be ''concave' ...

or vice versa.

# Notation (details)

## Leibniz's notation

The symbols $dx$, $dy$, and $\frac$ were introduced by
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, ', "having learned much"; Latin Latin (, or , ...
in 1675. It is still commonly used when the equation is viewed as a functional relationship between
dependent and independent variables Dependent and Independent variables are Variable and attribute (research), variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studi ...
. Then the first derivative is denoted by : $\frac,\quad\frac, \text\fracf,$ and was once thought of as an
infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. It also can be used to write the chain rule 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 Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaprime 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 A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...
, 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathemat ...
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 Arc length is the distance between two points along a section of a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. In ...

. It is typically used in
differential equation In mathematics, a differential equation is an equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

s in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

and
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
. 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 Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

's notation uses a
differential operator 300px, A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differential operator. In mathematics, a differential operator is an Operator (mathe ...
$D$, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " raise ...

and
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

ic 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

'': *: $\frac\sin\left(x\right) = \cos\left(x\right).$ *: $\frac\cos\left(x\right) = -\sin\left(x\right).$ *: $\frac\tan\left(x\right) = \sec^2\left(x\right) = \frac = 1 + \tan^2\left(x\right).$ * ''
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 Domain of a fun ...
'': *:

## 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 (mathematics), functions. For two functions, it may be stated in Notation for differentiatio ...
'': *: $\left(fg\right)\text{'} = f \text{'}g + fg\text{'}$ 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 f(x)=g(x)/h(x), where both and are differentiable and h(x)\neq 0. The quotient rule states that the deriv ...
'': *: $\left\left(\frac \right\right)\text{'} = \frac$ for all functions ''f'' and ''g'' at all inputs where . * '' Chain rule'' 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 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 (mathematics), functions. For two functions, it may be stated in Notation for differentiatio ...
. 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 extension of the real numbers, the derivative of a real function at a real point can be defined as the
shadow A shadow is a dark area where light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that can be visual perception, perceived by the human eye. Visible light is usually defined as havi ...
of the quotient for
infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
, 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 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 (geometric), vector, called the Differential geometry of curves, tangent vector, 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 of differentiation, linearity 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 The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

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 ∂ 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 Mass measured in grams is a function from this collection of weight to positive number">positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Converging lens or positive lens, i ...
. 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. It follows that the directional derivative is linear map, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 norm (mathematics), 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 (differential), pushforward 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 Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
. This matrix is called the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix (mathematics), matrix of all its first-order partial derivatives. When this matrix is square matrix, square, that is, when the function t ...
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 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 (mathematics), 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, 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 functor, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the function at that point. * An important generalization of the derivative concerns complex functions of Complex number, 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 – see holomorphic functions. * Another generalization concerns functions between smooth manifold, 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: the prototypical example is a smooth surface 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 bundles of ''M'' and ''N''. This definition is fundamental in
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
and has many uses – see pushforward (differential) and pullback (differential geometry). * Differentiation can also be defined for maps between Dimension (vector space), infinite dimensional vector spaces such as Banach spaces and Fréchet spaces. There is a generalization both of the directional derivative, called the Gateaux derivative, and of the differential, called the Fréchet derivative. * 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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
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 distribution (mathematics), 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. * The discrete equivalent of differentiation is finite differences. The study of differential calculus is unified with the calculus of finite differences in time scale calculus. * Also see arithmetic derivative.

# History

Calculus, known in its early history as ''infinitesimal calculus'', is a mathematics, mathematical discipline focused on limit (mathematics), limits, function (mathematics), functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Leibniz independently discovered calculus in the mid-17th century. However, each inventor claimed the other stole his work in a Leibniz–Newton calculus controversy, bitter dispute that continued until the end of their lives.

* Differential calculus#Applications of derivatives, Applications of derivatives * Automatic differentiation * Differentiability class * Differentiation rules * Differintegral * Fractal derivative * Generalizations of the derivative * Hasse derivative * History of calculus * Integral * Infinitesimal * Linearization * Mathematical analysis * Multiplicative inverse * Numerical differentiation * Rate (mathematics) * Radon–Nikodym theorem * Symmetric derivative * Schwarzian derivative

# Bibliography

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* * * * * * * * *

## Online books

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