In differential calculus, there is no single standard notation for differentiation. Instead, several notations for the derivative of a function or a dependent variable have been proposed by various mathematicians, including Leibniz, Newton, Lagrange, and Arbogast. 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, tensor analysis, or vector calculus—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) are listed below.
Leibniz's notation
The original notation employed by Gottfried Leibniz 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:
Furthermore, the derivative of at is therefore written
Higher derivatives are written as:
This is a suggestive notational device that comes from formal manipulations of symbols, as in,
The value of the derivative of at a point may be expressed in two ways using Leibniz's notation:
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 easy to remember and recognize:
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 infinitesimals. Later authors have assigned them other meanings, such as infinitesimals in non-standard analysis, or exterior derivatives. Commonly, is left undefined or equated with , while is assigned a meaning in terms of , via the equation
which may also be written as, for example,
(see below). Such equations give rise to the terminology found in some texts wherein the derivative is referred to as the "differential coefficient" (the coefficient of ).
Some authors and journals set the differential symbol in roman type instead of italic: . The ISO/IEC 80000 scientific style guide recommends this style.
Lagrange's notation
One of the most common modern notations for differentiation is named after Joseph Louis Lagrange, although it was in fact invented by Euler and popularized by the former. In Lagrange's notation, a prime mark denotes a derivative – hence it is sometimes called prime notation. If f is a function, then its derivative evaluated at x is written
It first appeared in print in 1749.
Higher derivatives are indicated using additional prime marks, as in for the second derivative and 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
to denote fourth, fifth, sixth, and higher order derivatives. Other authors use Arabic numerals in parentheses, as in
This notation also makes it possible to describe the nth derivative, where n is a variable. This is written
Unicode characters related to Lagrange's notation include
When there are two independent variables for a function , the following notation is sometimes used:The Differential and Integral Calculus (Augustus De Morgan, 1842). pp. 267-268
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
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 for the first integral (this is easily confused with the inverse function ), for the second integral, for the third integral, and for the nth integral.
D-notation
This notation is sometimes called although it was introduced by Louis François Antoine Arbogast, and it seems that Leonhard Euler did not use it.
This notation uses a differential operator 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
Higher derivatives are notated as "powers" of D (where the superscripts denote iterated composition of D), as in for the second derivative, for the third derivative, and for the nth 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 for the first derivative, for the second derivative, for the third derivative, and for the nth 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 are:
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 isWeisstein, Eric W. "Repeated Integral." From MathWorld--A Wolfram Web Resource. as follows for a first antiderivative, for a second antiderivative, and for an nth 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
Higher derivatives are represented using multiple dots, as in
Newton extended this idea quite far:Newton's notation reproduced from:
1st to 5th derivatives: Quadratura curvarum (Newton, 1704), p. 7 (p. 5r in original MS: ). 1st to 7th, nth and (n+1)th derivatives: Method of Fluxions (Newton, 1736), pp. 313-318 and p. 265 (p. 163 in original MS: ) 1st to 5th derivatives : A Treatise of Fluxions (Colin MacLaurin, 1742), p. 613 1st to 4th and nth derivatives: Articles "Differential" and "Fluxion", Dictionary of Pure and Mixed Mathematics (Peter Barlow, 1814) 1st to 4th, 10th and nth derivatives: Articles 622, 580 and 579 in A History of Mathematical Notations (F .Cajori, 1929) 1st to 6th and nth derivatives: The Mathematical Papers of Isaac Newton Vol. 7 1691-1695 (D. T. Whiteside, 1976), pp.88 and 17 1st to 3rd and nth derivatives: A History of Analysis (Hans Niels Jahnke, 2000), pp. 84-85 The dot for nth derivative may be omitted ( )
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 denotes velocityWeisstein, Eric W. "Overdot." From MathWorld--A Wolfram Web Resource. and denotes acceleration.Weisstein, Eric W. "Double Dot." From MathWorld--A Wolfram Web Resource. This notation is popular in physics and mathematical physics. 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:Article 580 in Florian Cajori, A History of Mathematical Notations (1929), Dover Publications, Inc. New York.
Newton developed the following partial differential operators using side-dots on a curved X ( ⵋ ). Definitions given by Whiteside are below:"Patterns of Mathematical Thought in the Later Seventeenth Century", Archive for History of Exact Sciences Vol. 1, No. 3 (D. T. Whiteside, 1961), pp. 361-362,378S.B. Engelsman has given more strict definitions in Families of Curves and the Origins of Partial Differentiation (2000), pp. 223-226
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).
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).
Higher order time integrals were as follows:Newton's notation for integration reproduced from: 1st to 3rd integrals: Quadratura curvarum (Newton, 1704), p. 7 (p. 5r in original MS: ) 1st to 3rd integrals: Method of Fluxions (Newton, 1736), pp. 265-266 (p. 163 in original MS: ) 4th integrals: The Doctrine of Fluxions (James Hodgson, 1736), pp. 54 and 72 1st to 2nd integrals: Articles 622 and 365 in A History of Mathematical Notations (F .Cajori, 1929) The nth integral notation is deducted from the nth derivative. It could be used in Methodus Incrementorum Directa & Inversa (Brook Taylor, 1715)
This mathematical notation did not become widespread because of printing difficulties and the Leibniz–Newton calculus controversy.
Partial derivatives
When more specific types of differentiation are necessary, such as in multivariate calculus or tensor analysis, 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:
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:
What makes this distinction important is that a non-partial derivative such as may, depending on the context, be interpreted as a rate of change in relative to when all variables are allowed to vary simultaneously, whereas with a partial derivative such as 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. The symbol is the derivative of the temperature T with respect to the volume V while keeping constant the entropy (subscript) S, while 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
and so on. Mixed partial derivatives can be expressed as
In this last case the variables are written in inverse order between the two notations, explained as follows:
So-called multi-index notation is used in situations when the above notation becomes cumbersome or insufficiently expressive. When considering functions on , we define a multi-index to be an ordered list of non-negative integers: . We then define, for , the notation
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