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 (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

The function whose value at x is 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,

Logically speaking, these equalities are not theorems. Instead, they are simply definitions of notation.

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 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.

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