Editing Differential calculus (section)

==History of differentiation==
{{Main|History of calculus}}

The concept of a derivative in the sense of a [[tangent line]] is a very old one, familiar to ancient [[Ancient Greece|Greek]] mathematicians such as [[Euclid]] (c. 300 BC), [[Archimedes]] (c. 287–212 BC), and [[Apollonius of Perga]] (c. 262–190 BC).<ref>See [[Euclid's Elements]], The [[Archimedes Palimpsest]] and {{MacTutor Biography|id=Apollonius|title=Apollonius of Perga}}</ref> [[Archimedes]] also made use of [[Cavalieri's principle|indivisibles]], although these were primarily used to study areas and volumes rather than derivatives and tangents (see ''[[The Method of Mechanical Theorems]]''). 
The use of infinitesimals to compute rates of change was developed significantly by [[Bhāskara II]] (1114–1185); indeed, it has been argued<ref>Ian G. Pearce. [http://turnbull.mcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch8_5.html Bhaskaracharya II.] {{Webarchive|url=https://web.archive.org/web/20160901092504/http://turnbull.mcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch8_5.html |date=2016-09-01 }}</ref> that many of the key notions of differential calculus can be found in his work, such as "[[Rolle's theorem]]".<ref>{{Cite journal|first1=T. A. A.|last1=Broadbent|title=Reviewed work(s): ''The History of Ancient Indian Mathematics'' by C. N. Srinivasiengar|journal=The Mathematical Gazette|volume=52|issue=381|date=October 1968|pages=307–8|doi=10.2307/3614212|jstor=3614212|last2=Kline|first2=M.|s2cid=176660647 }}</ref>

The mathematician, [[Sharaf al-Dīn al-Tūsī]] (1135–1213), in his ''Treatise on Equations'', established conditions for some cubic equations to have solutions, by finding the maxima of appropriate cubic polynomials. He obtained, for example, that the maximum (for positive {{mvar|x}}) of the cubic {{math| ''ax''<sup>2</sup> – ''x''<sup>3</sup>}} occurs when {{math|''x'' {{=}} 2''a'' / 3}}, and concluded therefrom that the equation {{math| ''ax''<sup>2</sup> {{=}} ''x''<sup>3</sup> + c}} has exactly one positive solution when {{math|''c'' {{=}} 4''a''<sup>3</sup> / 27}}, and two positive solutions whenever {{math|0 < ''c'' < 4''a''<sup>3</sup> / 27}}.{{sfn|Berggren|1990|page=307}} The historian of science, [[Roshdi Rashed]],{{sfn|Berggren|1990|page=308}} has argued that al-Tūsī must have used the derivative of the cubic to obtain this result. Rashed's conclusion has been contested by other scholars, however, who argue that he could have obtained the result by other methods which do not require the derivative of the function to be known.{{sfn|Berggren|1990|pp=308–309}}

The modern development of calculus is usually credited to [[Isaac Newton]] (1643–1727) and [[Gottfried Wilhelm Leibniz]] (1646–1716), who provided independent{{efn|Newton began his work in 1665 and Leibniz began his in 1676. However, Leibniz published his first paper in 1684, predating Newton's publication in 1693. It is possible that Leibniz saw drafts of Newton's work in 1673 or 1676, or that Newton made use of Leibniz's work to refine his own. Both Newton and Leibniz claimed that the other plagiarized their respective works. This resulted in a bitter [[Newton-Leibniz calculus controversy|controversy between them]] over who first invented calculus, which shook the mathematical community in the early 18th century.}} and unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the [[fundamental theorem of calculus]] relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes.{{efn|This was a monumental achievement, even though a restricted version had been proven previously by [[James Gregory (astronomer and mathematician)|James Gregory]] (1638–1675), and some key examples can be found in the work of [[Pierre de Fermat]] (1601–1665).}} For their ideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such as [[Pierre de Fermat]] (1607-1665), [[Isaac Barrow]] (1630–1677), [[René Descartes]] (1596–1650), [[Christiaan Huygens]] (1629–1695), [[Blaise Pascal]] (1623–1662) and [[John Wallis]] (1616–1703). Regarding Fermat's influence, Newton once wrote in a letter that "''I had the hint of this method [of fluxions] from Fermat's way of drawing tangents, and by applying it to abstract equations, directly and invertedly, I made it general.''"<ref name=Sabra>{{cite book | last = Sabra | first = A I.|title=Theories of Light: From Descartes to Newton | publisher = Cambridge University Press | year = 1981 | page = 144 | isbn = 978-0521284363}}</ref> Isaac Barrow is generally given credit for the early development of the derivative.<ref>Eves, H. (1990).</ref> Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to [[theoretical physics]], while Leibniz systematically developed much of the notation still used today.

Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as [[Augustin Louis Cauchy]] (1789–1857), [[Bernhard Riemann]] (1826–1866), and [[Karl Weierstrass]] (1815–1897). It was also during this period that the differentiation was generalized to [[Euclidean space]] and the [[complex plane]].

The 20th century brought two major steps towards our present understanding and practice of derivation : [[Lebesgue integration]], besides extending integral calculus to many more functions, clarified the relation between derivation and integration with the notion of [[absolute continuity]]. Later the [[theory of distributions]] (after [[Laurent Schwartz]]) extended derivation to generalized functions (e.g., the [[Dirac delta function]] previously introduced in [[Quantum Mechanics]]) and became fundamental to nowadays applied analysis especially by the use of [[weak solution]]s to [[partial differential equations]].