Editing Leibniz's notation (section)

{{Short description|Mathematical notation used for calculus}}
{{image frame|width=250|innerstyle=font-size:400%; line-height: 100%; font-family:Times New Roman, serif;display:flex;justify-content:space-evenly;align-items:center;|
caption = The first and second derivatives of <var>y</var> with respect to <var>x</var>, in the Leibniz notation. |
content = 
<div style="display:inline-block"><div style="border-bottom:2px solid black;padding-bottom:6px">''dy''</div><div>''dx''</div></div><div style="display:inline-block"><div style="border-bottom:2px solid black;padding-bottom:6px">''d''<sup>2</sup>''y''</div><div>''dx''<sup>2</sup></div></div>
}}
[[File:Gottfried Wilhelm Leibniz c1700.jpg|thumb|250px|[[Gottfried Wilhelm Leibniz|Gottfried Wilhelm von Leibniz]] (1646–1716), German philosopher, mathematician, and namesake of this widely used mathematical notation in calculus.]]

In [[calculus]], '''Leibniz's notation''', named in honor of the 17th-century German [[philosophy|philosopher]] and [[mathematician]] [[Gottfried Wilhelm Leibniz]], uses the symbols {{math|''dx''}} and {{math|''dy''}} to represent infinitely small (or [[infinitesimal]]) increments of {{math|''x''}} and {{math|''y''}}, respectively, just as {{math|Δ''x''}} and {{math|Δ''y''}} represent finite increments of {{math|''x''}} and {{math|''y''}}, respectively.<ref>{{cite book | last=Stewart | first=James | author-link=James Stewart (mathematician) | title=Calculus: Early Transcendentals | publisher=[[Brooks/Cole]] | edition=6th | year=2008 | isbn=978-0-495-01166-8 | url=https://archive.org/details/calculusearlytra00stew_1 }}</ref>

Consider {{math|''y''}} as a [[function (mathematics)|function]] of a variable {{math|''x''}}, or {{math|''y''}} = {{math|''f''(''x'')}}. If this is the case, then the [[derivative (mathematics)|derivative]] of {{math|''y''}} with respect to {{math|''x''}}, which later came to be viewed as the [[Limit (mathematics)|limit]]

:<math>\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x} = \lim_{\Delta x\rightarrow 0}\frac{f(x + \Delta x)-f(x)}{\Delta x},</math>

was, according to Leibniz, the [[quotient]] of an infinitesimal increment of {{math|''y''}} by an infinitesimal increment of {{math|''x''}}, or

:<math>\frac{dy}{dx}=f'(x),</math>

where the right hand side is [[Notation for differentiation#Lagrange's notation|Joseph-Louis Lagrange's notation]] for the derivative of {{math|''f''}} at {{math|''x''}}.  The infinitesimal increments are called {{em|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 kept it popular even in the face of several competing notations.  Several different formalisms were developed in the 20th century that can give rigorous meaning to notions of infinitesimals and infinitesimal displacements, including [[nonstandard analysis]], [[tangent space]], [[O notation]] and others.

The derivatives and integrals of calculus can be packaged into the modern theory of [[differential forms]], in which the derivative is genuinely a ratio of two differentials, and the integral likewise behaves in exact accordance with Leibniz notation.  However, this requires that derivative and integral first be defined by other means, and as such expresses the self-consistency and computational efficacy of the Leibniz notation rather than giving it a new foundation.