Notation for differentiation


In differential calculus, there is no single uniform notation for differentiation. Instead, several different notations for the derivative of a function or variable have been proposed by different 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 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
The function whose value at is 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,
Logically speaking, these equalities are not theorems. Instead, they are simply definitions of notation.
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. 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 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 or exterior derivatives.
Some authors and journals set the differential symbol in roman type instead of italic:. The ISO/IEC 80000 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. It is now the standard symbol for integration.

Lagrange's notation

One of the most common modern notations for differentiation is due to Joseph Louis Lagrange. In Lagrange's notation, a prime mark denotes a derivative. If f is a function, then its derivative evaluated at x is written
Lagrange first used the notation in unpublished works, and it appeared in print in 1770.
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 f, the following convention may be followed:

Lagrange's notation for antidifferentiation

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

Euler's notation

's notation uses a differential operator suggested by Louis François Antoine Arbogast, denoted as or When applied to a function, it is defined by
Higher derivatives are notated as powers of D, as in
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
When f is a function of several variables, it's common to use a "∂" rather than. As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function are:
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

Newton's notation

's 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:
Unicode characters related to Newton's notation include:
Newton's notation is generally used when the independent variable denotes time. If location is a function of t, then denotes velocity and denotes acceleration. This notation is popular in physics and mathematical physics. It also appears in areas of mathematics connected with physics such as differential equations. It is only popular for first and second derivatives, but in applications these are usually the only derivatives that are necessary.
When taking the derivative of a dependent variable y = f, an alternative notation exists:
Newton developed the following partial differential operators using side-dots on a curved X. Definitions given by Whiteside are below:

Newton's notation for integration

Newton developed many different notations for integration in his Quadratura curvarum 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.
To denote multiple integrals, Newton used two small vertical bars or primes, or a combination of previous symbols , to denote the second time integral.
Higher order time integrals were as follows:
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, 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 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
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:

Notation in vector calculus

concerns differentiation and integration 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, that A is a vector field with components, and that is a scalar field.
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.
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.