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| In [[calculus]], '''Leibniz's notation''', named in honor of the 17th-century [[Germany|German]] [[philosophy|philosopher]] and [[mathematics|mathematician]] [[Gottfried Leibniz|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''.<ref>{{cite book | last=Stewart | first=James | authorlink=James Stewart (mathematician) | title=Calculus: Early Transcendentals |publisher=[[Brooks/Cole]] | edition=6th | year=2008 | isbn=0-495-01166-5}}</ref> For ''y'' as a [[function (mathematics)|function]] of ''x'', or
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| :<math>y=f(x) \,,</math>
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| the [[derivative (mathematics)|derivative]] of ''y'' with respect to ''x'', which later came to be viewed as
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| :<math>\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x} = \lim_{\Delta x\rightarrow 0}\frac{f(x + \Delta x)-f(x)}{(x + \Delta x)-x},</math>
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| was, according to Leibniz, the [[quotient]] of an infinitesimal increment of ''y'' by an infinitesimal increment of ''x'', or
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| :<math>\frac{dy}{dx}=f'(x),</math>
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| where the right hand side is [[Notation for differentiation|Lagrange's notation]] for the derivative of ''f'' at ''x''. From the point of view of modern infinitesimal theory, <math>\Delta x</math> is an infinitesimal ''x''-increment, <math>\Delta y</math> is the corresponding ''y''-increment, and the derivative is the [[standard part]] of the infinitesimal ratio:
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| :<math>f'(x)={\rm st}\Bigg( \frac{\Delta y}{\Delta x} \Bigg)</math>.
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| Then one sets <math>dx=\Delta x</math>, <math>dy = f'(x) dx\,</math>, so that by definition, <math>f'(x)\,</math> is the ratio of ''dy'' by ''dx''.
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| Similarly, although mathematicians sometimes now view an integral
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| :<math>\int f(x)\,dx</math>
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| as a limit
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| :<math>\lim_{\Delta x\rightarrow 0}\sum_{i} f(x_i)\,\Delta x,</math>
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| where Δ''x'' is an interval containing ''x''<sub>''i''</sub>, 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.
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| ==History==
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| The Newton-Leibniz approach to [[infinitesimal calculus]] was introduced in the 17th century. While Newton did not have a standard notation for integration, Leibniz began using the <math>\int</math> character. He based the character on the Latin word ''summa'' ("sum"), which he wrote ''ſumma'' with the [[long s|elongated s]] commonly used in Germany at the time. This use first appeared publicly in his paper ''De Geometria'', published in ''[[Acta Eruditorum]]'' of June 1686,<ref>''Mathematics and its History'', John Stillwell, Springer 1989, p. 110</ref> but he had been using it in private manuscripts at least since 1675.<ref>''Early Mathematical Manuscripts of Leibniz'', J. M. Child, Open Court Publishing Co., 1920, pp. 73–74, 80.</ref>
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| In the 19th century, mathematicians ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians felt that the concept of [[infinitesimal]]s contained logical contradictions in its development. A number of 19th century mathematicians ([[Cauchy]], [[Weierstrass]] and others) found logically rigorous ways to treat derivatives and integrals without infinitesimals using limits as shown above. Nonetheless, Leibniz's notation is still in general use. Although the notation need not be taken literally, it is usually simpler than alternatives when the technique of [[separation of variables]] is used in the solution of differential equations. In physical applications, one may for example regard ''f''(''x'') as measured in meters per second, and d''x'' in seconds, so that ''f''(''x'') d''x'' is in meters, and so is the value of its definite integral. In that way the Leibniz notation is in harmony with [[dimensional analysis]].
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| In the 1960s, building upon earlier work by [[Edwin Hewitt]] and [[Jerzy Łoś]], [[Abraham Robinson]] developed rigorous mathematical explanations for Leibniz' intuitive notion of the "infinitesimal," and developed [[non-standard analysis]] based on these ideas. Robinson's methods are used by only a minority of mathematicians. [[Jerome Keisler]] wrote a [[Elementary calculus: an infinitesimal approach|first-year-calculus textbook]] based on Robinson's approach.
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| ==Leibniz's notation for differentiation==
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| In Leibniz's [[Mathematical notation|notation]] for [[derivative|differentiation]], the derivative of the function ''f''(''x'') is written:
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| :<math>\frac{d\bigl(f(x)\bigr)}{dx}\,.</math>
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| If we have a [[Variable (mathematics)|variable]] representing a function, for example if we set
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| :<math>y=f(x) \,,</math>
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| then we can write the derivative as:
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| :<math>\frac{dy}{dx}\,.</math>
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| Using [[Notation for differentiation|Lagrange's notation]], we can write:
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| :<math>\frac{d\bigl(f(x)\bigr)}{dx} = f'(x)\,.</math>
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| Using [[Newton's notation]], we can write:
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| :<math>\frac{dx}{dt} = \dot{x}\,.</math> | |
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| For higher derivatives, we express them as follows:
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| :<math>\frac{d^n\bigl(f(x)\bigr)}{dx^n}\text{ or }\frac{d^ny}{dx^n}</math>
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| denotes the ''n''th derivative of ƒ(''x'') or ''y'' respectively. Historically, this came from the fact that, for example, the third derivative is:
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| :<math>\frac{d \left(\frac{d \left( \frac{d \left(f(x)\right)} {dx}\right)} {dx}\right)} {dx}\,,</math>
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| which we can loosely write as:
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| :<math>\left(\frac{d}{dx}\right)^3 \bigl(f(x)\bigr) =
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| \frac{d^3}{\left(dx\right)^3} \bigl(f(x)\bigr)\,.</math>
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| Now drop the parentheses and we have:
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| :<math>\frac{d^3}{dx^3}\bigl(f(x)\bigr)\ \mbox{or}\ \frac{d^3y}{dx^3}\,.</math>
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| The [[chain rule]] and [[integration by substitution]] rules are especially easy to express here, because the "''d''" terms appear to cancel:
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| :<math>\frac{dy}{dx} = \frac{dy}{du_1} \cdot \frac{du_1}{du_2} \cdot \frac{du_2}{du_3}\cdots \frac{du_n}{dx}\,,</math> | |
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| etc., and:
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| :<math>\int y \, dx = \int y \frac{dx}{du} \, du.</math>
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| ==See also==
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| *[[Notation for differentiation]]
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| *[[Newton's notation]]
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| *[[Leibniz and Newton calculus controversy]]
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| ==Notes==
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| {{reflist}}
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| {{Infinitesimals}}
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| [[Category:Differential calculus]]
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| [[Category:Gottfried Leibniz]]
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| [[Category:History of calculus]]
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| [[Category:Mathematical notation]]
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| [[Category:Non-standard analysis]]
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| [[Category:Mathematics of infinitesimals]]
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