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In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols Template:Math and Template:Math to represent infinitely small (or infinitesimal) increments of Template:Math and Template:Math, respectively, just as Template:Math and Template:Math represent finite increments of Template:Math and Template:Math, respectively.<ref>Template:Cite book</ref>
Consider Template:Math as a function of a variable Template:Math, or Template:Math = Template:Math. If this is the case, then the derivative of Template:Math with respect to Template:Math, which later came to be viewed as the 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 Template:Math by an infinitesimal increment of Template:Math, or
- <math>\frac{dy}{dx}=f'(x),</math>
where the right hand side is Joseph-Louis Lagrange's notation for the derivative of Template:Math at Template:Math. The infinitesimal increments are called Template:Em. 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.
HistoryEdit
The Newton–Leibniz approach to infinitesimal calculus was introduced in the 17th century. While Newton worked with fluxions and fluents, Leibniz based his approach on generalizations of sums and differences.<ref name="Katz524">Template:Harvnb</ref> Leibniz adapted the integral symbol <math>\textstyle \int</math> from the initial elongated s of the Latin word Template:Serifumma ("sum") as written at the time. Viewing differences as the inverse operation of summation,<ref>Template:Harvnb</ref> he used the symbol Template:Mvar, the first letter of the Latin differentia, to indicate this inverse operation.<ref name="Katz524" /> Leibniz was fastidious about notation, having spent years experimenting, adjusting, rejecting and corresponding with other mathematicians about them.<ref>Template:Harvnb</ref> Notations he used for the differential of Template:Mvar ranged successively from Template:Mvar, Template:Mvar, and Template:Math until he finally settled on Template:Mvar.<ref>Template:Harvnb</ref> His integral sign first appeared publicly in the article "De Geometria Recondita et analysi indivisibilium atque infinitorum" ("On a hidden geometry and analysis of indivisibles and infinites"), published in Acta Eruditorum in June 1686,<ref>Template:Citation</ref><ref>Template:Cite book</ref> but he had been using it in private manuscripts at least since 1675.<ref>Template:Cite book</ref><ref>Leibniz, G. W., Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674-1676, Berlin: Akademie Verlag, 2008, pp. 288–295 Template:Webarchive ("Analyseos tetragonisticae pars secunda", October 29, 1675) and 321–331 ("Methodi tangentium inversae exempla", November 11, 1675).</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Leibniz first used Template:Mvar in the article "Nova Methodus pro Maximis et Minimis" also published in Acta Eruditorum in 1684.<ref name="Cajori204">Template:Harvnb</ref> While the symbol Template:Math does appear in private manuscripts of 1675,<ref>Leibniz, G. W., Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674-1676, Berlin: Akademie Verlag, 2008, pp. 321–331 esp. 328 ("Methodi tangentium inversae exempla", November 11, 1675).</ref><ref>Template:Harvnb</ref> it does not appear in this form in either of the above-mentioned published works. Leibniz did, however, use forms such as Template:Mvar and Template:Math in print.<ref name="Cajori204" />
At the end of the 19th century, Weierstrass's followers ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians felt that the concept of infinitesimals contained logical contradictions in its development. A number of 19th century mathematicians (Weierstrass and others) found logically rigorous ways to treat derivatives and integrals without infinitesimals using limits as shown above, while Cauchy exploited both infinitesimals and limits (see Cours d'Analyse). 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 dx in seconds, so that f(x) dx is in meters, and so is the value of its definite integral. In that way the Leibniz notation is in harmony with dimensional analysis.
Leibniz's notation for differentiationEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Suppose a dependent variable Template:Math represents a function Template:Math of an independent variable Template:Math, that is,
- <math>y=f(x).</math>
Then the derivative of the function Template:Math, in Leibniz's notation for differentiation, can be written as
- <math>\frac{dy}{dx}\,\text{ or }\frac{d}{dx}y\,\text{ or }\frac{d\bigl(f(x)\bigr)}{dx}.</math>
The Leibniz expression, also, at times, written Template:Math, is one of several notations used for derivatives and derived functions. A common alternative is Lagrange's notation
- <math>\frac{dy}{dx}\, = y' = f'(x).</math>
Another alternative is Newton's notation, often used for derivatives with respect to time (like velocity), which requires placing a dot over the dependent variable (in this case, Template:Math):
- <math>\frac{dx}{dt} = \dot{x}.</math>
Lagrange's "prime" notation is especially useful in discussions of derived functions and has the advantage of having a natural way of denoting the value of the derived function at a specific value. However, the Leibniz notation has other virtues that have kept it popular through the years.
In its modern interpretation, the expression Template:Math should not be read as the division of two quantities Template:Mvar and Template:Mvar (as Leibniz had envisioned it); rather, the whole expression should be seen as a single symbol that is shorthand for
- <math>\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}</math>
(note Template:Math vs. Template:Mvar, where Template:Math indicates a finite difference).
The expression may also be thought of as the application of the differential operator Template:Math (again, a single symbol) to Template:Mvar, regarded as a function of Template:Mvar. This operator is written Template:Mvar in Euler's notation. Leibniz did not use this form, but his use of the symbol Template:Mvar corresponds fairly closely to this modern concept.
While there is traditionally no division implied by the notation (but see Nonstandard analysis), the division-like notation is useful since in many situations, the derivative operator does behave like a division, making some results about derivatives easy to obtain and remember.<ref>Template:Cite book</ref> This notation owes its longevity to the fact that it seems to reach to the very heart of the geometrical and mechanical applications of the calculus.<ref>Template:Harvnb</ref>
Leibniz notation for higher derivativesEdit
If Template:Math, the Template:Mvarth derivative of Template:Mvar in Leibniz notation is given by,<ref name="Briggs">Template:Harvnb</ref>
- <math>f^{(n)}(x) = \frac{d^ny}{dx^n}.</math>
This notation, for the second derivative, is obtained by using Template:Math as an operator in the following way,<ref name="Briggs"/>
- <math>\frac{d^2y}{dx^2} \,=\, \frac{d}{dx}\left(\frac{dy}{dx}\right).</math>
A third derivative, which might be written as,
- <math>\frac{d \left(\frac{d \left( \frac{dy}{dx}\right)}{dx}\right)}{dx}\,,</math>
can be obtained from
- <math>\frac{d^3y}{dx^3} \,=\, \frac{d}{dx}\left(\frac{d^2y}{dx^2}\right) \,=\, \frac{d}{dx}\left( \frac{d}{dx}\left(\frac{dy}{dx}\right)\right).</math>
Similarly, the higher derivatives may be obtained inductively.
While it is possible, with carefully chosen definitions, to interpret Template:Math as a quotient of differentials, this should not be done with the higher order forms.<ref>Template:Harvnb</ref> However, an alternative Leibniz notation for differentiation for higher orders allows for this.Template:Cn
This notation was, however, not used by Leibniz. In print he did not use multi-tiered notation nor numerical exponents (before 1695). To write Template:Math for instance, he would write Template:Mvar, as was common in his time. The square of a differential, as it might appear in an arc length formula for instance, was written as Template:Mvar. However, Leibniz did use his Template:Mvar notation as we would today use operators, namely he would write a second derivative as Template:Mvar and a third derivative as Template:Mvar. In 1695 Leibniz started to write Template:Math and Template:Math for Template:Mvar and Template:Mvar respectively, but l'Hôpital, in his textbook on calculus written around the same time, used Leibniz's original forms.<ref>Template:Harvnb</ref>
Leibniz's notation for integrationEdit
Leibniz introduced the integral symbol for integration<ref>Earliest Uses of Symbols of Calculus", School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 1 May 2025.</ref> (or "antidifferentiation") now commonly used today:<math>\displaystyle \int</math>
The notation was introduced in 1675 in his private writings;<ref>Gottfried Wilhelm Leibniz, Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674–1676, Berlin: Akademie Verlag, 2008, pp. 288–295 Template:Webarchive ("Analyseos tetragonisticae pars secunda", October 29, 1675) and 321–331 Template:Webarchive ("Methodi tangentium inversae exempla", November 11, 1675).</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> it first appeared publicly in the article "{{#invoke:Lang|lang}}" (On a hidden geometry and analysis of indivisibles and infinites), published in Acta Eruditorum in June 1686.<ref>Template:Citation</ref><ref>Template:Cite book</ref> The symbol was based on the ſ (long s) character and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal summands.
Use in various formulasEdit
One reason that Leibniz's notations in calculus have endured so long is that they permit the easy recall of the appropriate formulas used for differentiation and integration. For instance, the chain rule—suppose that the function Template:Mvar is differentiable at Template:Mvar and Template:Math is differentiable at Template:Math. Then the composite function Template:Math is differentiable at Template:Mvar and its derivative can be expressed in Leibniz notation as,<ref>Template:Harvnb</ref>
- <math>\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.</math>
This can be generalized to deal with the composites of several appropriately defined and related functions, Template:Math and would be expressed as,
- <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>
Also, the integration by substitution formula may be expressed by<ref>Template:Harvnb</ref>
- <math>\int y \, dx = \int y \frac{dx}{du} \, du,</math>
where Template:Mvar is thought of as a function of a new variable Template:Mvar and the function Template:Mvar on the left is expressed in terms of Template:Mvar while on the right it is expressed in terms of Template:Mvar.
If Template:Math where Template:Mvar is a differentiable function that is invertible, the derivative of the inverse function, if it exists, can be given by,<ref>Template:Harvnb</ref>
- <math>\frac{dx}{dy} = \frac{1}{\left( \frac{dy}{dx} \right)},</math>
where the parentheses are added to emphasize the fact that the derivative is not a fraction.
However, when solving differential equations, it is easy to think of the Template:Maths and Template:Maths as separable. One of the simplest types of differential equations is<ref>Template:Harvnb</ref>
- <math>M(x) + N(y) \frac{dy}{dx} = 0,</math>
where Template:Mvar and Template:Mvar are continuous functions. Solving (implicitly) such an equation can be done by examining the equation in its differential form,
- <math>M(x) dx + N(y) dy = 0</math>
and integrating to obtain
- <math>\int M(x) \, dx + \int N(y) \, dy = C.</math>
Rewriting, when possible, a differential equation into this form and applying the above argument is known as the separation of variables technique for solving such equations.
In each of these instances the Leibniz notation for a derivative appears to act like a fraction, even though, in its modern interpretation, it isn't one.
Modern justification of infinitesimalsEdit
In the 1960s, building upon earlier work by Edwin Hewitt and Jerzy Łoś, Abraham Robinson developed mathematical explanations for Leibniz's infinitesimals that were acceptable by contemporary standards of rigor, and developed nonstandard analysis based on these ideas. Robinson's methods are used by only a minority of mathematicians. Jerome Keisler wrote a first-year calculus textbook, Elementary calculus: an infinitesimal approach, based on Robinson's approach.
From the point of view of modern infinitesimal theory, Template:Math is an infinitesimal Template:Math-increment, Template:Math is the corresponding Template:Math-increment, and the derivative is the standard part of the infinitesimal ratio:
- <math>f'(x)={\rm st}\Bigg( \frac{\Delta y}{\Delta x} \Bigg)</math>.
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 Template:Math by Template:Math.
Similarly, although most mathematicians now view an integral
- <math>\int f(x)\,dx</math>
as a limit
- <math>\lim_{\Delta x\rightarrow 0}\sum_{i} f(x_i)\,\Delta x,</math>
where Template:Math is an interval containing Template:Math, Leibniz viewed it as the sum (the integral sign denoted summation for him) of infinitely many infinitesimal quantities Template:Math. From the viewpoint of nonstandard analysis, it is correct to view the integral as the standard part of such an infinite sum.
The trade-off needed to gain the precision of these concepts is that the set of real numbers must be extended to the set of hyperreal numbers.
Other notations of LeibnizEdit
Leibniz experimented with many different notations in various areas of mathematics. He felt that good notation was fundamental in the pursuit of mathematics. In a letter to l'Hôpital in 1693 he says:<ref name="Cajori185">Template:Harvnb</ref> Template:Quote He refined his criteria for good notation over time and came to realize the value of "adopting symbolisms which could be set up in a line like ordinary type, without the need of widening the spaces between lines to make room for symbols with sprawling parts."<ref>Template:Harvnb</ref> For instance, in his early works he heavily used a vinculum to indicate grouping of symbols, but later he introduced the idea of using pairs of parentheses for this purpose, thus appeasing the typesetters who no longer had to widen the spaces between lines on a page and making the pages look more attractive.<ref>Template:Harvnb</ref>
Many of the over 200 new symbols introduced by Leibniz are still in use today.<ref>Template:Harvnb</ref> Besides the differentials Template:Mvar, Template:Mvar and the integral sign ( ∫ ) already mentioned, he also introduced the colon (:) for division, the middle dot (⋅) for multiplication, the geometric signs for similar (~) and congruence (≅), the use of Recorde's equal sign (=) for proportions (replacing Oughtred's :: notation) and the double-suffixTemplate:Cfn notation for determinants.<ref name="Cajori185" />