differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...

, there is no single standard notation for differentiation. Instead, several notations for the

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

of a function or a

dependent variable A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical functio ...

have been proposed by various mathematicians, including

Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many ...

, Newton,

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaArbogast. The usefulness of each notation depends on the context in which it is used, and it is sometimes advantageous to use more than one notation in a given context. For more specialized settings—such as partial derivatives in

multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables ('' mult ...

tensor analysis In mathematics and physics, a tensor field is a function (mathematics), function assigning a tensor to each point of a region (mathematics), region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tens ...

, or

vector calculus Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...

—other notations, such as subscript notation or the ∇ operator are common. The most common notations for differentiation (and its opposite operation, antidifferentiation or

indefinite integration In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated s ...

) are listed below.

Leibniz's notation

The original notation employed by

Gottfried Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ...

is used throughout mathematics. It is particularly common when the equation is regarded as a functional relationship between

dependent and independent variables A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function ...

and . Leibniz's notation makes this relationship explicit by writing the derivative as:

\frac.

Furthermore, the derivative of at is therefore written

\frac(x)\text\frac\text\frac f(x).

Higher derivatives are written as:

\frac, \frac, \frac, \ldots, \frac.

This is a suggestive notational device that comes from formal manipulations of symbols, as in,

\frac = \left(\frac\right)^2y = \frac.

The value of the derivative of at a point may be expressed in two ways using Leibniz's notation:

\left.\frac\_ \text \frac(a).

Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially helpful when considering

partial derivative 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). P ...

s. It also makes the

chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, 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 ...

easy to remember and recognize:

\frac = \frac \cdot \frac.

Leibniz's notation for differentiation does not require assigning meaning to symbols such as or (known as differentials) on their own, and some authors do not attempt to assign these symbols meaning. Leibniz treated these symbols as

infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...

s. Later authors have assigned them other meanings, such as infinitesimals in

non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...

, or

exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...

s. Commonly, is left undefined or equated with

\Delta x

, while is assigned a meaning in terms of , via the equation :

dy = \frac \cdot dx,

which may also be written, e.g. :

df = f'(x) \cdot dx

(see below). Such equations give rise to the terminology found in some texts wherein the derivative is referred to as the "differential coefficient" (i.e., the

coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...

of ). Some authors and journals set the differential symbol in

roman type In Latin script typography, roman is one of the three main kinds of Typeface, historical type, alongside blackletter and Italic type, italic. Sometimes called normal or regular, it is distinct from these two for its upright style (relative to the ...

instead of italic: . The

ISO/IEC 80000 ISO/IEC 80000, ''Quantities and units'', is an international standard describing the International System of Quantities (ISQ). It was developed and promulgated jointly by the International Organization for Standardization (ISO) and the Intern ...

scientific style guide recommends this style.

Lagrange's notation

One of the most common modern notations for differentiation is named after

Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaEuler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

and popularized by the former. In Lagrange's notation, a prime mark denotes a derivative. If ''f'' is a function, then its derivative evaluated at ''x'' is written :

f'(x)

. It first appeared in print in 1749. Higher derivatives are indicated using additional prime marks, as in

f''(x)

for the

second derivative In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the secon ...

and

f(x)

for the

third derivative In calculus, a branch of mathematics, the third derivative or third-order derivative is the rate at which the second derivative, or the rate of change of the rate of change, is changing. The third derivative of a function y = f(x) can be denot ...

. The use of repeated prime marks eventually becomes unwieldy; some authors continue by employing

Roman numeral Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, ea ...

s, usually in lower case, as in :

f^(x), f^(x), f^(x), \ldots,

to denote fourth, fifth, sixth, and higher order derivatives. Other authors use Arabic numerals in parentheses, as in :

f^(x), f^(x), f^(x), \ldots.

This notation also makes it possible to describe the ''n''th derivative, where ''n'' is a variable. This is written :

f^(x).

Unicode characters related to Lagrange's notation include * * * * When there are two independent variables for a function

f(x,y)

, the following notation was sometimes used:''The Differential and Integral Calculus'' (

Augustus De Morgan Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He is best known for De Morgan's laws, relating logical conjunction, disjunction, and negation, and for coining the term "mathematical induction", the ...

, 1842). pp. 267-268 :

f_ &= \frac = f_ \end

Lagrange's notation for antidifferentiation

When taking the antiderivative, Lagrange followed Leibniz's notation:

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiainverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...

f^(x)

), :

f^(x)

for the second integral, :

f^(x)

for the third integral, and :

f^(x)

for the ''n''th integral.

D-notation

This notation is sometimes called although it was introduced by Louis François Antoine Arbogast, and it seems that

Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

did not use it. This notation uses a

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

denoted as (D operator) or (Newton–Leibniz operator).Weisstein, Eric W. "Differential Operator." From ''MathWorld''--A Wolfram Web Resource. When applied to a function , it is defined by :

(Df)(x) = \frac.

Higher derivatives are notated as "powers" of ''D'' (where the superscripts denote iterated

composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...

of ''D''), as in :

D^2f

for the second derivative, :

D^3f

for the third derivative, and :

D^nf

for the ''n''th derivative. D-notation leaves implicit the variable with respect to which differentiation is being done. However, this variable can also be made explicit by putting its name as a subscript: if ''f'' is a function of a variable ''x'', this is done by writing :

D_x f

for the first derivative, :

D^2_x f

for the second derivative, :

D^3_x f

for the third derivative, and :

D^n_x f

for the ''n''th derivative. When ''f'' is a function of several variables, it is common to use " ∂", a stylized cursive lower-case d, rather than "". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function

f(x,y)

are: :

& \partial_ f = \frac. \end

See . D-notation is useful in the study of differential equations and in differential algebra.

D-notation for antiderivatives

D-notation can be used for antiderivatives in the same way that Lagrange's notation is as follows :

D^f(x)

for a first antiderivative, :

D^f(x)

for a second antiderivative, and :

D^f(x)

for an ''n''th antiderivative.

Newton's notation

Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...

's notation for differentiation (also called the dot notation, fluxions, or sometimes, crudely, the flyspeck notation for differentiation) places a dot over the dependent variable. That is, if ''y'' is a function of ''t'', then the derivative of ''y'' with respect to ''t'' is :

\dot y

Higher derivatives are represented using multiple dots, as in :

\ddot y, \overset

Newton extended this idea quite far: :

\overset &\equiv \frac = D_t^n y = f^(t) = y^_t \end

Unicode characters related to Newton's notation include: * * * ← replaced by "combining diaeresis" + "combining dot above". * ← replaced by "combining diaeresis" twice. * * * * * Newton's notation is generally used when the independent variable denotes

time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...

. If location is a function of ''t'', then

\dot y

denotes

velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...

and

\ddot y

denotes

acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...

. This notation is popular in

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

and

mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...

. It also appears in areas of mathematics connected with physics such as differential equations. When taking the derivative of a dependent variable ''y'' = ''f''(''x''), an alternative notation exists: :

\frac = \dot:\dot \equiv \frac:\frac = \frac = \frac = \frac\Bigl(f(x)\Bigr) = D y = f'(x) = y'.

Newton developed the following partial differential operators using side-dots on a curved X ( ⵋ ). Definitions given by Whiteside are below: :

\cdot\mathcal\!\cdot\ \ &=\ xy\frac = xy f_\,, \end

Newton's notation for integration

Newton developed many different notations for integration in his ''Quadratura curvarum'' (1704) and later works: he wrote a small vertical bar or prime above the dependent variable ( ), a prefixing rectangle (), or the inclosure of the term in a rectangle () to denote the '' fluent'' or time integral ( absement). :

\begin
                      y &= \Box \dot \equiv \int \dot \,dt = \int f'(t) \,dt = D_t^ (D_t y) = f(t) + C_0 = y_t + C_0 \\
  \overset &= \Box y \equiv \int y \,dt = \int f(t) \,dt = D_t^ y = F(t) + C_1
\end

To denote multiple integrals, Newton used two small vertical bars or primes (), or a combination of previous symbols , to denote the second time integral (absity). :

\overset = \Box \overset \equiv \int \overset \,dt = \int F(t) \,dt = D_t^ y = g(t) + C_2

Higher order time integrals were as follows: :

\begin
        \overset &= \Box \overset \equiv \int \overset \,dt = \int g(t) \,dt = D_t^ y = G(t) + C_3 \\
  \overset &= \Box \overset \equiv \int \overset \,dt = \int G(t) \,dt = D_t^ y = h(t) + C_4 \\
       \overset\overset &= \Box \overset\oversety \equiv \int \overset\oversety \,dt = \int s(t) \,dt = D_t^ y = S(t) + C_n
\end

This

mathematical notation Mathematical notation consists of using glossary of mathematical symbols, symbols for representing operation (mathematics), operations, unspecified numbers, relation (mathematics), relations, and any other mathematical objects and assembling ...

did not become widespread because of printing difficulties^{'Citation needed''/sup> and the Leibniz–Newton calculus controversy.

Partial derivatives

When more specific types of differentiation are necessary, such as in multivariate calculus

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables ('' mult ...
or tensor analysis

In mathematics and physics, a tensor field is a function (mathematics), function assigning a tensor to each point of a region (mathematics), region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tens ...
, other notations are common.

For a function ''f'' of a single independent variable ''x'', we can express the derivative using subscripts of the independent variable:

: $f_ &= \frac. \end$

This type of notation is especially useful for taking partial derivative

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). P ...
s of a function of several variables.

Partial derivatives are generally distinguished from ordinary derivatives by replacing the differential operator ''d'' with a " ∂" symbol. For example, we can indicate the partial derivative of with respect to ''x'', but not to ''y'' or ''z'' in several ways:
: $\frac = f_x = \partial_x f.$
What makes this distinction important is that a non-partial derivative such as $\textstyle \frac$ ''may'', depending on the context, be interpreted as a rate of change in $f$ relative to $x$ when all variables are allowed to vary simultaneously, whereas with a partial derivative such as $\textstyle \frac$ it is explicit that only one variable should vary.

Other notations can be found in various subfields of mathematics, physics, and engineering; see for example the Maxwell relations of thermodynamics

Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
. The symbol $\left(\frac\right)_$ is the derivative of the temperature ''T'' with respect to the volume ''V'' while keeping constant the entropy (subscript) ''S'', while $\left(\frac\right)_$ is the derivative of the temperature with respect to the volume while keeping constant the pressure ''P''. This becomes necessary in situations where the number of variables exceeds the degrees of freedom, so that one has to choose which other variables are to be kept fixed.

Higher-order partial derivatives with respect to one variable are expressed as
: $& \frac = f_, \end$
and so on. Mixed partial derivatives can be expressed as

: $\frac = f_.$

In this last case the variables are written in inverse order between the two notations, explained as follows:

: $& \frac\!\left(\frac\right) = \frac. \end$
So-called multi-index notation is used in situations when the above notation becomes cumbersome or insufficiently expressive. When considering functions on $\R^n$ , we define a multi-index to be an ordered list of $n$ non-negative integers: $\alpha = (\alpha_1,\ldots,\alpha_n), \ \alpha_i \in \Z_$ . We then define, for $f:\R^n \to X$ , the notation

: $\partial^\alpha f = \frac \cdots \frac f$

In this way some results (such as the Leibniz rule) that are tedious to write in other ways can be expressed succinctly -- some examples can be found in the article on multi-indices.

Notation in vector calculus

Vector calculus

Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...
concerns differentiation and integration of vector

Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism

Vector may also refer to:
Mathematics a ...
or scalar fields. Several notations specific to the case of three-dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
are common.

Assume that is a given Cartesian coordinate system

In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
, that A is a vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
with components $\mathbf = (A_x, A_y, A_z)$ , and that $\varphi = \varphi(x,y,z)$ is a scalar field

In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
.

The differential operator introduced by William Rowan Hamilton

Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...
, written ∇ and called del

Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
or nabla, is symbolically defined in the form of a vector,
: $\nabla = \left( \frac, \frac, \frac \right)\!,$
where the terminology ''symbolically'' reflects that the operator ∇ will also be treated as an ordinary vector.

* Gradient

In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
: The gradient $\mathrm \varphi$ of the scalar field $\varphi$ is a vector, which is symbolically expressed by the multiplication

Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
of ∇ and scalar field '' $\varphi$ '',

:: $\begin
\operatorname \varphi
&= \left( \frac, \frac, \frac \right) \\
&= \left( \frac, \frac, \frac \right) \varphi \\
&= \nabla \varphi
\end$

* Divergence

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
: The divergence $\mathrm\,\mathbf$ of the vector field A is a scalar, which is symbolically expressed by the dot product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
of ∇ and the vector A,

:: $\begin
\operatorname \mathbf
&= + + \\
&= \left( \frac, \frac, \frac \right) \cdot \mathbf \\
&= \nabla \cdot \mathbf
\end$

* Laplacian

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
: The Laplacian $\operatorname \operatorname \varphi$ of the scalar field $\varphi$ is a scalar, which is symbolically expressed by the scalar multiplication of ∇² and the scalar field ''φ'',

:: $\begin
\operatorname \operatorname \varphi
&= \nabla \cdot (\nabla \varphi) \\
&= (\nabla \cdot \nabla) \varphi \\
&= \nabla^2 \varphi \\
&= \Delta \varphi \\
\end$

* Rotation

Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
: The rotation $\mathrm\,\mathbf$ , or $\mathrm\,\mathbf$ , of the vector field A is a vector, which is symbolically expressed by the cross product

In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of ∇ and the vector A,

:: $\begin
\operatorname \mathbf
&= \left(
- ,
- ,
-
\right) \\
&= \left( - \right) \mathbf +
\left( - \right) \mathbf +
\left( - \right) \mathbf \\
&= \begin
\mathbf & \mathbf & \mathbf \\
\cfrac & \cfrac & \cfrac \\
A_x & A_y & A_z
\end \\
&= \nabla \times \mathbf
\end$

Many symbolic operations of derivatives can be generalized in a straightforward manner by the gradient operator in Cartesian coordinates. For example, the single-variable product rule

In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
has a direct analogue in the multiplication of scalar fields by applying the gradient operator, as in

: $(f g)' = f' g+f g' ~~~ \Longrightarrow ~~~ \nabla(\phi \psi) = (\nabla \phi) \psi + \phi (\nabla \psi).$

Many other rules from single variable calculus have vector calculus analogues for the gradient, divergence, curl, and Laplacian.

Further notations have been developed for more exotic types of spaces. For calculations in Minkowski space

In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.

The model helps show how a ...
, the d'Alembert operator, also called the d'Alembertian, wave operator, or box operator is represented as $\Box$ , or as $\Delta$ when not in conflict with the symbol for the Laplacian.

See also

*
*
*
*
*
*
* Operational calculus
Operational calculus, also known as operational analysis, is a technique by which problems in Mathematical Analysis, analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomia ...

References

External links

Earliest Uses of Symbols of Calculus
maintained by Jeff Miller ().

{{Differential equations topics

Differential calculus
Mathematical notation}