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

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 mat ...

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

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 ...

s. Later authors have assigned them other meanings, such as infinitesimals in non-standard analysis 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 Electrotech ...

scientific style guide recommends this style.

Leibniz's notation for antidifferentiation

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

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, ...

. :

\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 and

f(x)

for the third derivative. The use of repeated prime marks eventually becomes unwieldy. Some authors continue by employing Roman numerals, 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 ...

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 ...

's notation uses a differential operator suggested by Louis François Antoine Arbogast, 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 ∂ (Unicode: U+2202) is a stylized cursive '' d'' mainly used as a mathematical symbol, usually to denote a partial derivative such as / (read as "the partial derivative of ''z'' with respect to ''x''"). It is also used for the boun ...

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

\dot y

denotes velocity 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 the ...

. This notation is popular in physics and mathematical physics. 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

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 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 file:Thermodynamic map.svg, 400px, Flow chart showing the paths between the Maxwell relations. P is pressure, T temperature, V volume, S entropy, \alpha coefficient of thermal expansion, \kappa compressibility, C_V heat capacity at constant volu ...

of thermodynamics. 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 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 Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following: * Product rule in differential calculus * General Leibniz rule, a generalization of the product rule * Leibniz integral rule * The alternating series test, al ...

) 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 concerns differentiation and

of vector or scalar fields. Several notations specific to the case of three-dimensional Euclidean space are common. Assume that is a given

Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...

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

. The differential operator introduced by William Rowan Hamilton, 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 t ...

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: 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 algeb ...

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 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 is d ...

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, 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 M ...

, 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