In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

of a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

variable Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...

have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation (and its opposite operation, the antidifferentiation or indefinite integration) are listed below.

Leibniz's notation

The original notation employed by

Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...

is used throughout mathematics. It is particularly common when the equation is regarded as a functional relationship between dependent and independent variables 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 derivatives. It also makes 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 , ...

easy to remember and recognize: :

\frac = \frac \cdot \frac.

Leibniz's notation for differentiation does not require assigning a meaning to symbols such as or on their own, and some authors do not attempt to assign these symbols meaning. Leibniz treated these symbols as infinitesimals. 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 epsilon–delta ...

or exterior derivatives. Some authors and journals set the differential symbol in roman type instead of italic: . The

ISO/IEC 80000 ISO 80000 or IEC 80000 is an international standard introducing the International System of Quantities (ISQ). It was developed and promulgated jointly by the International Organization for Standardization (ISO) and the International Electrote ...

scientific style guide recommends this style.

Leibniz's notation for antidifferentiation

Leibniz introduced the

integral symbol The integral symbol: : (Unicode), \displaystyle \int (LaTeX) is used to denote integrals and antiderivatives in mathematics, especially in calculus. History The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz in 1 ...

in ''Analyseos tetragonisticae pars secunda'' and ''Methodi tangentium inversae exempla'' (both from 1675). It is now the standard symbol for integration. :

\begin
                                                   \int y'\,dx &= \int f'(x)\,dx = f(x) + C_0 = y + C_0 \\
                                                    \int y\,dx &= \int f(x)\,dx = F(x) + C_1 \\
                                             \iint y\,dx^2 &= \int \left ( \int y\,dx \right ) dx = \int_ f(x)\,dx = \int F(x)\,dx = g(x) + C_2 \\
  \underbrace_ y\,\underbrace_n &= \int_ f(x)\,dx = \int s(x)\,dx = S(x) + C_n
\end

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 and just 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 . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...

and

f(x)

for the

third derivative In calculus, a branch of mathematics, the third 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 denoted by :\frac,\quad f(x),\qu ...

. 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 convention may be followed:''The Differential and Integral Calculus'' ( Augustus De Morgan, 1842). pp. 267-268 :

\begin
          f^\prime &= \frac = f_x \\
          f_\prime &= \frac = f_y \\
  f^ &= \frac = f_ \\
   f_\prime^\prime &= \frac\ = f_ \\
  f_ &= \frac = f_
\end

Lagrange's notation for antidifferentiation

When taking the antiderivative, Lagrange followed Leibniz's notation: Lagrange, ''Nouvelle méthode pour résoudre les équations littérales par le moyen des séries'' (1770), p. 25-26. http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN308900308, LOG_0017&physid=PHYS_0031 :

f(x) = \int f'(x)\,dx = \int y\,dx.

However, because integration is the inverse operation of differentiation, Lagrange's notation for higher order derivatives extends to integrals as well. Repeated integrals of ''f'' may be written as :

f^(x)

for the first integral (this is easily confused with the

inverse 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 X ...

f^(x)

), :

f^(x)

for the second integral, :

f^(x)

for the third integral, and :

f^(x)

for the ''n''th integral.

Euler's notation

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

's notation uses a differential operator suggested by

Louis François Antoine Arbogast Louis François Antoine Arbogast (4 October 1759 – 8 April 1803) was a French mathematician. He was born at Mutzig in Alsace and died at Strasbourg, where he was professor. He wrote on series and the derivatives known by his name: he was the ...

, 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 v ...

of ''D''), as in :

D^2f

for the second derivative, :

D^3f

for the third derivative, and :

D^nf

for the ''n''th derivative. Euler's notation leaves implicit the variable with respect to which differentiation is being done. However, this variable can also be notated explicitly. When ''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's common to use "

∂ The Character (symbol), character ∂ (Unicode: U+2202) is a stylized cursive ''d'' mainly used as a Table of mathematical symbols, mathematical symbol, usually to denote a partial derivative such as / (read as "the partial derivative of ''z'' wit ...

", 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 are: :

\partial_ f = \frac,

\partial_ f = \frac,

\partial_ f = \frac,

\partial_ f = \frac.

See . Euler's notation is useful for stating and solving

linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...

s, as it simplifies presentation of the differential equation, which can make seeing the essential elements of the problem easier.

Euler's notation for antidifferentiation

Euler's notation can be used for antidifferentiation 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 (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...

'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: :

\begin
                 \ddot &\equiv \frac = \frac\left(\frac\right)  = \frac\Bigl(\dot\Bigr) = \frac\Bigl(f'(t)\Bigr) = D_t^2 y = f''(t) = y''_t \\
         \overset &= \dot \equiv \frac = D_t^3 y = f(t) = y_t \\
   \overset &= \overset = \ddot \equiv \frac = D_t^4 y = f^(t) = y^_t \\
   \overset &= \ddot = \dot = \ddot \equiv \frac = D_t^5 y = f^(t) = y^_t \\
   \overset &= \overset \equiv \frac = D_t^6 y = f^(t) = y^_t \\
   \overset &= \dot \equiv \frac = D_t^7 y = f^(t) = y^_t \\
  \overset &= \ddot \equiv \frac = D_t^ y = f^(t) = y^_t \\
   \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 continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...

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

\dot y

denotes

velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...

and

\ddot y

denotes

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 t ...

. This notation is popular in

physics Physics is the natural science that studies matter, its 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 which r ...

and

mathematical physics Mathematical physics refers to the development of 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 developme ...

. It also appears in areas of mathematics connected with physics such as

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. 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: :

\begin
       \mathcal                                                            \ &=\  f(x,y) \,, \\
       \cdot\mathcal                                                       \ &=\  x\frac = xf_x\,, \\
       \mathcal\!\cdot                                                     \ &=\  y\frac = yf_y\,, \\
       \colon\!\mathcal\,\text\,\cdot\!\left(\cdot\mathcal\right) \ &=\  x^2\frac = x^2 f_\,, \\
       \mathcal\colon\,\text\,\left(\mathcal\cdot\right)\!\cdot   \ &=\  y^2\frac = y^2 f_\,, \\
       \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 In kinematics, absement (or absition) is a measure of sustained displacement of an object from its initial position, i.e. a measure of how far away and for how long. The word ''absement'' is a portmanteau of the words ''absence'' and ''displ ...

). :

\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 symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathem ...

did not become widespread because of printing difficulties and the

Leibniz–Newton calculus controversy In the history of calculus, the calculus controversy (german: Prioritätsstreit, lit=priority dispute) was an argument between the mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first invented calculus. The question was a ...

Partial derivatives

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

tensor analysis In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...

, other notations are common. For a function ''f'' of an independent variable ''x'', we can express the derivative using subscripts of the independent variable: :

\begin
      f_x &= \frac \\
  f_ &= \frac.
\end

This type of notation is especially useful for taking partial derivatives 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, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...

. 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_,

\frac = f_,

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: :

(f_)_ = f_,

\frac\!\left(\frac\right) = \frac.

So-called

multi-index notation Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...

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,..,\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 concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...

concerns differentiation and integration of

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

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, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

are common. Assume that is a given Cartesian coordinate system, that A is a vector field with components

\mathbf = (\mathbf_x, \mathbf_y, \mathbf_z)

, and that

\varphi = \varphi(x,y,z)

is a scalar field. The differential operator introduced by

William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irela ...

, 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 of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

: The gradient

\mathrm \varphi

of the scalar field

\varphi

is a vector, which is symbolically expressed by the multiplication 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 quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...

: 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 as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...

of ∇ and the vector A, ::

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

* Laplacian: 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: The rotation

\mathrm\,\mathbf

, or

\mathrm\,\mathbf

, of the vector field A is a vector, which is symbolically expressed by the cross product 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 mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...

, the

d'Alembert operator In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...

, 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.

References

External links

Earliest Uses of Symbols of Calculus
maintained by Jeff Miller (). {{Differential equations topics Differential calculus Mathematical notation