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, just as Δx and Δy represent finite increments of x and y. For y as a function of x, or

the derivative of y with respect to x, which later came to be viewed as

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 Lagrange's notation for the derivative of f at x. From the point of view of modern infinitesimal theory, is an infinitesimal x-increment, is the corresponding y-increment, and the derivative is the standard part of the infinitesimal ratio:

.

Then one sets, so that by definition, is the ratio of dy by dx.

Similarly, although mathematicians sometimes now view an integral

as a limit

where Δx is an interval containing x_i, Leibniz viewed it as the sum (the integral sign denoting summation) of infinitely many infinitesimal quantities f(x) dx. From the modern viewpoint, it is more correct to view the integral as the standard part of an infinite sum of such quantities.