Leibniz's notation

In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively.

Consider y as a function of a variable x, or y = f(x). If this is the case, then the derivative of y with respect to x, which later came to be viewed as the limit

was, according to Leibniz, the quotient of an infinitesimal increment of y by an infinitesimal increment of x, or

where the right hand side is Joseph-Louis Lagrange's notation for the derivative of f at x.

Leibniz's concept of infinitesimals, long considered to be too imprecise to be used as a foundation of calculus, was eventually replaced by rigorous concepts developed by Weierstrass and others. Consequently, Leibniz's quotient notation was re-interpreted to stand for the limit of the modern definition. However, in many instances, the symbol did seem to act as an actual quotient would and its usefulness kept it popular even in the face of several competing notations. In the modern rigorous treatment of non-standard calculus, justification can be found to again consider the notation as representing an actual quotient.

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