Notation For Differentiation - Leibniz's Notation

Leibniz's Notation

See also: Leibniz's notation dy dx d 2y dx2

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. In this case the derivative can be written as:

The function whose value at x is the derivative of f at x is therefore written

(although strictly speaking this denotes the variable value of the derivative function rather than the derivative function itself).

Higher derivatives are expressed as

for the nth derivative of y = f(x). Historically, this came from the fact that, for example, the third derivative is:

which we can loosely write (dropping the brackets in the denominator) as:

as above.

With Leibniz's notation, the value of the derivative of y at a point x = a can be written in two different ways:

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:

In the formulation of calculus in terms of limits, the du symbol has been assigned various meanings by various authors.

Some authors do not assign a meaning to du by itself, but only as part of the symbol du/dx.

Others define dx as an independent variable, and use d(x + y) = dx + dy and d(x·y) = dx·y + x·dy as formal axioms for differentiation. See differential algebra.

In non-standard analysis du is defined as an infinitesimal.

It is also interpreted as the exterior derivative du of a function u.

See differential (infinitesimal) for further information.