function of a real variable
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
measures the sensitivity to change of the function value (output value) with respect to a change in its
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
(input value). Derivatives are a fundamental tool of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.
Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real fu ...
of several variables, the Jacobian matrix reduces to the gradient vector.
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 is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
''. The
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, o ...
relates antidifferentiation with
integration
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...
. Differentiation and integration constitute the two fundamental operations in single-variable calculus.
Definition
A
function of a real variable
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
is ''differentiable'' at a point of its domain, if its domain contains an
open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
containing , and the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
:
exists. This means that, for every positive real number (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 mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
(see
(ε, δ)-definition of limit
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
).
If the function is differentiable at , that is if the limit exists, then this limit is called the ''derivative'' of at , and denoted (read as " prime of ") or (read as "the derivative of with respect to at ", " by at ", or " over at "); see , below.
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 g ...
at . As an example, choose a point and let be the step function 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 mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
function given by is continuous at , but it is not differentiable there. If is positive, then the slope of the secant line from 0 to is one, whereas if is negative, then the slope of the secant line from 0 to is negative one. This can be seen graphically as a "kink" or a "cusp" in the graph at . Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by is not differentiable at .
In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.
Most functions that occur in practice have derivatives at all points or at
almost every
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 ...
point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points. Under mild conditions, for example if the function is a monotone function or a Lipschitz function, this is true. However, in 1872 Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function. In 1931, Stefan Banach proved that the set of functions that have a derivative at some point is a
meager set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
in the space of all continuous functions. Informally, this means that hardly any random continuous functions have a derivative at even one point.
Derivative as a function
Let be a function that has a derivative at every point in its domain. We can then define a function that maps every point to the value of the derivative of at . This function is written and is called the ''derivative function'' or the ''derivative of'' .
Sometimes has a derivative at most, but not all, points of its domain. The function whose value at equals whenever is defined and elsewhere is undefined is also called the derivative of . It is still a function, but its domain may be smaller than the domain of .
Using this idea, differentiation becomes a function of functions: The derivative is an
operator
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another s ...
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 '' second derivative of ''. Similarly, the derivative of the second derivative, if it exists, is written and is called the '' third derivative of ''. Continuing this process, one can define, if it exists, the th derivative as the derivative of the th derivative. These repeated derivatives are called ''higher-order derivatives''. The th derivative is also called the derivative of order and denoted .
If represents the position of an object at time , then the higher-order derivatives of have specific interpretations in physics. The first derivative of is the object's velocity. The second derivative of is the
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
. The third derivative of is the jerk. And finally, the fourth through sixth derivatives of are
snap, crackle, and pop
Snap, Crackle and Pop are the cartoon mascots of Rice Krispies, a brand of breakfast cereal marketed by Kellogg's.
History
The gnome characters were originally designed by illustrator Vernon Grant in the early 1930s. The names are onomatopoeia ...
; most applicable to
astrophysics
Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the he ...
.
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 mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
. (This is a stronger condition than having derivatives, as shown by the second example of .) A function that has infinitely many derivatives is called ''infinitely differentiable'' or ''
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
''.
On the real line, every polynomial function is infinitely differentiable. By standard differentiation rules, if a polynomial of degree is differentiated times, then it becomes a
constant function
In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic properti ...
. 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 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 to being a
concave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.
Definition
A real-valued function f on an i ...
or vice versa.
Notation (details)
Leibniz's notation
The symbols , , and were introduced by Gottfried Wilhelm Leibniz in 1675. It is still commonly used when the equation is viewed as a functional relationship between dependent and independent variables. Then the first derivative is denoted by
:
and was once thought of as an
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally ref ...
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, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Par ...
. It also can be used to write the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
as
:
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 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 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 may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
. It is typically used in
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s in physics and differential geometry. The dot notation, however, becomes unmanageable for high-order derivatives (order 4 or more) and cannot deal with multiple independent variables.
Euler's notation
Euler's notation uses a differential operator , 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 equations.
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, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
*Expo ...