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 Δ*x* and Δ*y* 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 Δ*x* 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.

Read more about Leibniz's Notation: History, Leibniz's Notation For Differentiation

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