In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable 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.
The first and second derivatives of y with respect to x, in the Leibniz notation.
The original notation employed by Gottfried Leibniz is used throughout mathematics. It is particularly common when the equation y = f(x) is regarded as a functional relationship between dependent and independent variablesy and x. Leibniz's notation makes this relationship explicit by writing the derivative as
Furthermore, the derivative of f at x 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 y at a point x = a 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 a meaning to symbols such as dx or dy 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.
Some authors and journals set the differential symbol d in roman type instead of italic: dx. The ISO/IEC 80000 scientific style guide recommends this style.
Leibniz's notation for antidifferentiationEdit
The single and double indefinite integrals of y with respect to x, in the Leibniz notation.
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.
A function f of x, differentiated once in Lagrange's notation.
One of the most common modern notations for differentiation is named after Joseph Louis Lagrange, even though it was actually invented by Euler 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
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
U+2033◌″DOUBLE PRIME (double derivative)
U+2034◌‴TRIPLE PRIME (third derivative)
U+2057◌⁗QUADRUPLE PRIME (fourth derivative)
When there are two independent variables for a function f(x, y), the following convention may be followed:
Lagrange's notation for antidifferentiationEdit
The single and double indefinite integrals of f with respect to x, in the Lagrange notation.
When taking the antiderivative, Lagrange followed Leibniz's notation:
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
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.
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
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's common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are:
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 antidifferentiationEdit
D−1 xy D−2f
The x antiderivative of y and the second antiderivative of f, Euler notation.
Euler's notation can be used for antidifferentiation in the same way that Lagrange's notation is as follows
for a first antiderivative,
for a second antiderivative, and
for an nth antiderivative.
The first and second derivatives of x, 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
When taking the derivative of a dependent variable y = f(x), an alternative notation exists:
Newton developed the following partial differential operators using side-dots on a curved X ( ⵋ ). Definitions given by Whiteside are below:
Newton's notation for integrationEdit
The first and second antiderivatives of x, in one of Newton's notations.
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 (y̍ ), a prefixing rectangle (▭y), or the inclosure of the term in a rectangle (y) to denote the fluent or time integral (absement).
To denote multiple integrals, Newton used two small vertical bars or primes (y̎), or a combination of previous symbols ▭y̍y̍, to denote the second time integral (absity).
For a function f of an 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.
A function f differentiated against x.
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 f(x, y, z) 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
The differential operator introduced by William Rowan Hamilton, written ∇ and called del or nabla, is symbolically defined in the form of a vector,
where the terminology symbolically reflects that the operator ∇ will also be treated as an ordinary vector.
Gradient of the scalar field φ.
Gradient: The gradient of the scalar field is a vector, which is symbolically expressed by the multiplication of ∇ and scalar field ,
The divergence of the vector field A.
Divergence: The divergence of the vector field A is a scalar, which is symbolically expressed by the dot product of ∇ and the vector A,
The Laplacian of the scalar field φ.
Laplacian: The Laplacian of the scalar field is a scalar, which is symbolically expressed by the scalar multiplication of ∇2 and the scalar field φ,
The curl of vector field A.
Rotation: The rotation , or , of the vector field A is a vector, which is symbolically expressed by the cross product of ∇ and the vector A,
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 has a direct analogue in the multiplication of scalar fields by applying the gradient operator, as in
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, also called the d'Alembertian, wave operator, or box operator is represented as , or as when not in conflict with the symbol for the Laplacian.
1st to 5th derivatives: Quadratura curvarum (Newton, 1704), p. 7 (p. 5r in original MS: "Newton Papers : On the Quadrature of Curves". Archived from the original on 2016-02-28. Retrieved 2016-02-05.).
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: "Newton Papers : Fluxions". Archived from the original on 2017-04-06. Retrieved 2016-02-05.)
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)