
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:
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:
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 ...
(even very small), there exists a positive real number
such that, for every such that
and
then
is defined, and
:
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
(read as " prime of ") or
(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
:
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
and
refer to corresponding changes, i.e.
:
.
The above formula holds because
:
Thus
:
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
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,
:
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 :
:
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:
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*** "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
:
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''
:
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 :
:
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 :
:
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:
:
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:
:
The operator , however, is not defined on individual numbers. It is only defined on functions:
:
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
:
Calculation shows that is a differentiable function whose derivative at
is given by
:
is twice the absolute value function at
, 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
:
in the sense that
:
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
, or it may fail to exist, as in the case of the inflection point of the function given by
. 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
,
, and
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
:
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
:
for the ''n''th derivative of
. These are abbreviations for multiple applications of the derivative operator. For example,
:
With Leibniz's notation, we can write the derivative of
at the point
in two different ways:
:
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
:
Lagrange's notation
Sometimes referred to as ''prime notation'', one of the most common modern notations for differentiation is due to
, so that the derivative of a function
is denoted
. Similarly, the second and third derivatives are denoted
:
and
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:
:
or
The latter notation generalizes to yield the notation
for the ''n''th derivative of
– 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
, then
:
and
denote, respectively, the first and second derivatives of
. 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 ...
, which is applied to a function
to give the first derivative
. The ''n''th derivative is denoted
.
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
:
or
,
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'':
*:
* ''
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'':
*:
*:
*:
*:
* ''
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 ...

'':
*:
*:
*:
* ''
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 ...
'':
*: