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 lim Δ x → 0 Δ y Δ x = lim Δ x → 0 f ( x + Δ x ) − f ( x ) Δ x , {\displaystyle \lim _{\Delta x\rightarrow 0}{\frac {\Delta y}{\Delta x}}=\lim _{\Delta x\rightarrow 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}},} was, according to Leibniz, the quotient of an infinitesimal increment of y by an infinitesimal increment of x, or d y d x = f ′ ( x ) , {\displaystyle {\frac {dy}{dx}}=f'(x),} where the right hand side is Joseph-Louis Lagrange's notation for the derivative of f at x. The infinitesimal increments are called differentials. Related to this is the integral in which the infinitesimal increments are summed (e.g. to compute lengths, areas and volumes as sums of tiny pieces), for which Leibniz also supplied a closely related notation involving the same differentials, a notation whose efficiency proved decisive in the development of continental European mathematics. 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 in the 19th century. 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...