Incalculus,Leibniz's notation, named in honor of the 17th-century Germanphilosopher andmathematicianGottfried Wilhelm Leibniz, uses the symbolsdx anddy to represent infinitely small (orinfinitesimal) increments ofx andy, respectively, just asΔx andΔy represent finite increments ofx andy, respectively.^[1]

Considery as afunction of a variablex, ory =f(x). If this is the case, then thederivative ofy with respect tox, which later came to be viewed as thelimit

${\displaystyle \underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x},}$

was, according to Leibniz, thequotient of an infinitesimal increment ofy by an infinitesimal increment ofx, or

${\displaystyle \frac{dy}{dx}=f^{\prime }(x),}$

where the right hand side isJoseph-Louis Lagrange's notation for the derivative off atx. The infinitesimal increments are calleddifferentials. Related to this is theintegral 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 byWeierstrass 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, includingnonstandard analysis,tangent space,O notation and others.

The derivatives and integrals of calculus can be packaged into the modern theory ofdifferential 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.

The Newton–Leibniz approach toinfinitesimal calculus was introduced in the 17th century. While Newton worked withfluxions and fluents, Leibniz based his approach on generalizations of sums and differences.^[2] Leibniz adapted theintegral symbol${\displaystyle \int }$ from the initialelongated s of the Latin wordſumma ("sum") as written at the time. Viewing differences as the inverse operation of summation,^[3] he used the symbold, the first letter of the Latindifferentia, to indicate this inverse operation.^[2] Leibniz was fastidious about notation, having spent years experimenting, adjusting, rejecting and corresponding with other mathematicians about them.^[4] Notations he used for the differential ofy ranged successively fromω,l, and⁠y/d⁠ until he finally settled ondy.^[5] Hisintegral 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 inActa Eruditorum in June 1686,^[6][7] but he had been using it in private manuscripts at least since 1675.^[8][9][10] Leibniz first useddx in the article "Nova Methodus pro Maximis et Minimis" also published inActa Eruditorum in 1684.^[11] While the symbol⁠dx/dy⁠ does appear in private manuscripts of 1675,^[12][13] it does not appear in this form in either of the above-mentioned published works. Leibniz did, however, use forms such asdy ad dx anddy :dx in print.^[11]

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 ofinfinitesimals 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 (seeCours 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 ofseparation of variables is used in the solution of differential equations. In physical applications, one may for example regardf(x) as measured in meters per second, and dx in seconds, so thatf(x) dx is in meters, and so is the value of its definite integral. In that way the Leibniz notation is in harmony withdimensional analysis.

Suppose adependent variabley represents a functionf of an independent variablex, that is,

${\displaystyle y=f(x).}$

Then the derivative of the functionf, in Leibniz'snotation fordifferentiation, can be written as

${\displaystyle \frac{dy}{dx}\text{ or }\frac{d}{dx}y\text{ or }\frac{d(f(x))}{dx}.}$

The Leibniz expression, also, at times, writtendy/dx, is one of several notations used for derivatives and derived functions. A common alternative isLagrange's notation

${\displaystyle \frac{dy}{dx}=y^{\prime }=f^{\prime }(x).}$

Another alternative isNewton's notation, often used for derivatives with respect to time (likevelocity), which requires placing a dot over the dependent variable (in this case,x):

${\displaystyle \frac{dx}{dt}=\dot{x}.}$

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 expression⁠dy/dx⁠ should not be read as the division of two quantitiesdx anddy (as Leibniz had envisioned it); rather, the whole expression should be seen as a single symbol that is shorthand for

${\displaystyle \underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}}$

(noteΔ vs.d, whereΔ indicates a finite difference).

The expression may also be thought of as the application of thedifferential operator⁠d/dx⁠ (again, a single symbol) toy, regarded as a function ofx. This operator is writtenD inEuler's notation. Leibniz did not use this form, but his use of the symbold corresponds fairly closely to this modern concept.

While there is traditionally no division implied by the notation (but seeNonstandard 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.^[14]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.^[15]

Ify =f(x), thenth derivative off in Leibniz notation is given by,^[16]

${\displaystyle f^{(n)}(x)=\frac{d^{n}y}{d{x}^n}.}$

This notation, for thesecond derivative, is obtained by using⁠d/dx⁠ as an operator in the following way,^[16]

${\displaystyle \frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right).}$

A third derivative, which might be written as,

${\displaystyle \frac{d\left(\frac{d\left(\frac{dy}{dx}\right)}{dx}\right)}{dx},}$

can be obtained from

${\displaystyle \frac{d^{3}y}{d{x}^3}=\frac{d}{dx}\left(\frac{d^{2}y}{d{x}^2}\right)=\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{dy}{dx}\right)\right).}$

Similarly, the higher derivatives may be obtained inductively.

While it is possible, with carefully chosen definitions, to interpret⁠dy/dx⁠ as a quotient ofdifferentials, this should not be done with the higher order forms.^[17] However, an alternative Leibniznotation for differentiation for higher orders allows for this.^{[citation needed]}

This notation was, however, not used by Leibniz. In print he did not use multi-tiered notation nor numerical exponents (before 1695). To writex³ for instance, he would writexxx, as was common in his time. The square of a differential, as it might appear in anarc length formula for instance, was written asdxdx. However, Leibniz did use hisd notation as we would today use operators, namely he would write a second derivative asddy and a third derivative asdddy. In 1695 Leibniz started to writed²⋅x andd³⋅x forddx anddddx respectively, butl'Hôpital, in his textbook on calculus written around the same time, used Leibniz's original forms.^[18]

Leibniz introduced theintegral symbol for integration^[19] (or "antidifferentiation") now commonly used today:${\displaystyle {\displaystyle \int }}$

The notation was introduced in 1675 in his private writings;^[20][21] it 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 inActa Eruditorum in June 1686.^[22][23] The symbol was based on the ſ (long s) character and was chosen because Leibniz thought of the integral as an infinitesum of infinitesimalsummands.

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, thechain rule—suppose that the functiong is differentiable atx andy =f(u) is differentiable atu =g(x). Then the composite functiony =f(g(x)) is differentiable atx and its derivative can be expressed in Leibniz notation as,^[24]

${\displaystyle \frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}.}$

This can be generalized to deal with the composites of several appropriately defined and related functions,u₁,u₂, ...,u_n and would be expressed as,

${\displaystyle \frac{dy}{dx}=\frac{dy}{d{u}_1}\cdot \frac{d{u}_1}{d{u}_2}\cdot \frac{d{u}_2}{d{u}_3}\cdots \frac{d{u}_n}{dx}.}$

Also, theintegration by substitution formula may be expressed by^[25]

${\displaystyle \int ydx=\int y\frac{dx}{du}du,}$

wherex is thought of as a function of a new variableu and the functiony on the left is expressed in terms ofx while on the right it is expressed in terms ofu.

Ify =f(x) wheref is a differentiable function that isinvertible, the derivative of the inverse function, if it exists, can be given by,^[26]

${\displaystyle \frac{dx}{dy}=\frac{1}{\left(\frac{dy}{dx}\right)},}$

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 thedys anddxs as separable. One of the simplest types ofdifferential equations is^[27]

${\displaystyle M(x)+N(y)\frac{dy}{dx}=0,}$

whereM andN are continuous functions. Solving (implicitly) such an equation can be done by examining the equation in itsdifferential form,

${\displaystyle M(x)dx+N(y)dy=0}$

and integrating to obtain

${\displaystyle \int M(x)dx+\int N(y)dy=C.}$

Rewriting, when possible, a differential equation into this form and applying the above argument is known as theseparation 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.

In the 1960s, building upon earlier work byEdwin Hewitt andJerzy Łoś,Abraham Robinson developed mathematical explanations for Leibniz's infinitesimals that were acceptable by contemporary standards of rigor, and developednonstandard 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,Δx is an infinitesimalx-increment,Δy is the correspondingy-increment, and the derivative is thestandard part of the infinitesimal ratio:

${\displaystyle f^{\prime }(x)=\mathrm{s}\mathrm{t}(\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x})}$.

Then one sets${\displaystyle dx=\mathrm{\Delta }x}$,${\displaystyle dy=f^{\prime }(x)dx}$, so that by definition,${\displaystyle f^{\prime }(x)}$ is the ratio ofdy bydx.

Similarly, although most mathematicians now view an integral

${\displaystyle \int f(x)dx}$

as a limit

${\displaystyle \underset{\mathrm{\Delta }x\to 0}{lim}\sum _{i}f(x_i)\mathrm{\Delta }x,}$

whereΔx is an interval containingx_i, Leibniz viewed it as the sum (the integral sign denoted summation for him) of infinitely many infinitesimal quantitiesf(x) dx. 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 ofreal numbers must be extended to the set ofhyperreal numbers.

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:^[28]

One of the secrets of analysis consists in the characteristic, that is, in the art of skilful employment of the available signs, and you will observe, Sir, by the small enclosure [on determinants] that Vieta and Descartes have not known all the mysteries.

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."^[29] For instance, in his early works he heavily used avinculum 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.^[30]

Many of the over 200 new symbols introduced by Leibniz are still in use today.^[31] Besides the differentialsdx,dy 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 ofRecorde's equal sign (=) for proportions (replacingOughtred's:: notation) and the double-suffix^{[clarification needed]} notation for determinants.^[28]

• Leibniz–Newton calculus controversy

• Briggs, William; Cochran, Lyle (2010),Calculus / Early Transcendentals / Single Variable, Addison-Wesley,ISBN 978-0-321-66414-3
• Cajori, Florian (1993) [1928],A History of Mathematical Notations, New York: Dover,ISBN 0-486-67766-4
• Katz, Victor J. (1993),A History of Mathematics / An Introduction (2nd ed.), Addison Wesley Longman,ISBN 978-0-321-01618-8
• Mazur, Joseph (2014),Enlightening Symbols / A Short History of Mathematical Notation and Its Hidden Powers, Princeton University Press,ISBN 978-0-691-17337-5
• Swokowski, Earl W. (1983),Calculus with Analytic Geometry (Alternate ed.), Prindle, Weber and Schmidt,ISBN 0-87150-341-7

• Differential calculus
• Gottfried Wilhelm Leibniz
• History of calculus
• Mathematical notation
• Nonstandard analysis
• Mathematics of infinitesimals

• Webarchive template wayback links
• Articles with short description
• Short description is different from Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from March 2024
• Articles containing Latin-language text
• Wikipedia articles needing clarification from October 2022