Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians.
The most common notations for differentiation (and its opposite operation, the antidifferentiation or indefinite integration) are listed below.
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 variables y and x. Leibniz's notation makes this relationship explicit by writing the derivative as:[1]
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 for differentiation does not require assigning meaning to symbols such as dx or dy (known as differentials) on their own, and some authors do not attempt to assign these symbols meaning.
Later authors have assigned them other meanings, such as infinitesimals in non-standard analysis, or exterior derivatives.
, while dy is assigned a meaning in terms of dx, via the equation which may also be written, e.g. (see below).
Some authors and journals set the differential symbol d in roman type instead of italic: dx.
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.
The use of repeated prime marks eventually becomes unwieldy.
Some authors continue by employing Roman numerals, usually in lower case,[4][5] 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(x, y), the following convention may be followed:[6] When taking the antiderivative, Lagrange followed Leibniz's notation:[7] 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 This notation is sometimes called Euler's notation although it was introduced by Louis François Antoine Arbogast,[8] and it seems that Leonhard Euler did not use it.
[10] When applied to a function f(x), it is defined by Higher derivatives are notated as "powers" of D (where the superscripts denote iterated composition of D), as in[6] D-notation leaves implicit the variable with respect to which differentiation is being done.
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:[13] Unicode characters related to Newton's notation include: Newton's notation is generally used when the independent variable denotes time.
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:[16] Newton developed the following partial differential operators using side-dots on a curved X ( ⵋ ).
Definitions given by Whiteside are below:[17][18] 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 ▭y̍ y̍, to denote the second time integral (absity).
Higher order time integrals were as follows:[19] This mathematical notation did not become widespread because of printing difficulties[Citation needed] and the Leibniz–Newton calculus controversy.
When more specific types of differentiation are necessary, such as in multivariate calculus or tensor analysis, other notations are common.
Other notations can be found in various subfields of mathematics, physics, and engineering; see for example the Maxwell relations of thermodynamics.
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.
Several notations specific to the case of three-dimensional Euclidean space are common.
Assume that (x, y, z) is a given Cartesian coordinate system, that A is a vector field with components
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.
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.