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Concept
Notation for differentiation
Summary
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.
Leibniz's 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 variables y 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 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.
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Related concepts (17)
Time derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as . A variety of notations are used to denote the time derivative. In addition to the normal (Leibniz's) notation, A very common short-hand notation used, especially in physics, is the 'over-dot'. I.E. (This is called Newton's notation) Higher time derivatives are also used: the second derivative with respect to time is written as with the corresponding shorthand of .
Differential of a function
In calculus, the differential represents the principal part of the change in a function with respect to changes in the independent variable. The differential is defined by where is the derivative of f with respect to , and is an additional real variable (so that is a function of and ). The notation is such that the equation holds, where the derivative is represented in the Leibniz notation , and this is consistent with regarding the derivative as the quotient of the differentials.
Second derivative
In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation: where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change.