Editing Differential (mathematics) (section)

== History and usage ==
{{See also|History of calculus}}

[[Infinitesimal]] quantities played a significant role in the development of calculus. [[Archimedes]] used them, even though he did not believe that arguments involving infinitesimals were rigorous.<ref>{{Harvnb|Boyer|1991}}.</ref> [[Isaac Newton]] referred to them as [[Method of Fluxions|fluxions]]. However, it was [[Gottfried Leibniz]] who coined the term ''differentials'' for infinitesimal quantities and introduced the notation for them which is still used today.

In [[Leibniz's notation]], if ''x'' is a variable quantity, then ''dx'' denotes an infinitesimal change in the variable ''x''. Thus, if ''y'' is a function of ''x'', then the [[derivative]] of ''y'' with respect to ''x'' is often denoted ''dy''/''dx'', which would otherwise be denoted (in the notation of Newton or [[Joseph-Louis Lagrange|Lagrange]]) ''ẏ'' or ''y''{{′}}. The use of differentials in this form attracted much criticism, for instance in the famous pamphlet [[The Analyst]] by Bishop Berkeley. Nevertheless, the notation has remained popular because it suggests strongly the idea that the derivative of ''y'' at ''x'' is its [[instantaneous rate of change]] (the [[slope (mathematics)|slope]] of the graph's [[tangent line]]), which may be obtained by taking the [[limit (mathematics)|limit]] of the ratio Δ''y''/Δ''x'' as Δ''x'' becomes arbitrarily small. Differentials are also compatible with [[dimensional analysis]], where a differential such as ''dx'' has the same dimensions as the variable ''x''.
	
Calculus evolved into a distinct branch of mathematics during the 17th century CE, although there were antecedents going back to antiquity. The presentations of, e.g., Newton, Leibniz, were marked by non-rigorous definitions of terms like differential, [[fluent (mathematics)|fluent]] and "infinitely small". While many of the arguments in [[George Berkeley|Bishop Berkeley]]'s 1734 [[The Analyst]] are theological in nature, modern mathematicians acknowledge the validity of his argument against "[[The Analyst#Ghosts of departed quantities|the Ghosts of departed Quantities]]"; however, the modern approaches do not have the same technical issues. Despite the lack of rigor, immense progress was made in the 17th and 18th centuries. In the 19th century, Cauchy and others gradually developed the [[Epsilon, delta]] approach to continuity, limits and derivatives, giving a solid conceptual foundation for calculus.
	
In the 20th century, several new concepts in, e.g., multivariable calculus, differential geometry, seemed to encapsulate the intent of the old terms, especially ''differential''; both differential and infinitesimal are used with new, more rigorous, meanings.

Differentials are also used in the notation for [[integral]]s because an integral can be regarded as an infinite sum of infinitesimal quantities: the area under a graph is obtained by subdividing the graph into infinitely thin strips and summing their areas. In an expression such as
<math display=block>\int f(x) \,dx,</math>
the integral sign (which is a modified [[long s]]) denotes the infinite sum, ''f''(''x'') denotes the "height" of a thin strip, and the differential ''dx'' denotes its infinitely thin width.