Editing Leibniz's notation (section)

==Leibniz's notation for differentiation==
{{main|Notation for differentiation}}
Suppose a [[Dependent and independent variables#Mathematics|dependent variable]] {{math|''y''}} represents a function {{math|''f''}} of an independent variable {{math|''x''}}, that is,

:<math>y=f(x).</math>

Then the derivative of the function {{math|''f''}}, in Leibniz's [[Mathematical notation|notation]] for [[derivative|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 {{math|''dy''/''dx''}}, 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 [[Notation for differentiation#Newton's notation|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, {{math|''x''}}):

:<math>\frac{dx}{dt} = \dot{x}.</math>

Lagrange's "[[Prime (symbol)|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 {{math|{{sfrac|''dy''|''dx''}}}} should not be read as the division of two quantities {{mvar|dx}} and {{mvar|dy}} (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 {{math|Δ}} vs. {{mvar|d}}, where {{math|Δ}} indicates a finite difference).

The expression may also be thought of as the application of the [[differential operator]] {{math|{{sfrac|''d''|''dx''}}}} (again, a single symbol) to {{mvar|y}}, regarded as a function of {{mvar|x}}. This operator is written {{mvar|D}} in [[Notation for differentiation#Euler's notation|Euler's notation]]. Leibniz did not use this form, but his use of the symbol {{mvar|d}} 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>{{cite book |first1=D. W. |last1=Jordan |first2=P. |last2=Smith |title=Mathematical Techniques: An Introduction for the Engineering, Physical, and Mathematical Sciences |publisher=Oxford University Press |year=2002 |page=58}}</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>{{harvnb|Cajori|1993|loc=Vol. II, p. 262}}</ref>

===Leibniz notation for higher derivatives===

If {{math|1=''y'' = ''f''(''x'')}}, the {{mvar|n}}th derivative of {{mvar|f}} in Leibniz notation is given by,<ref name="Briggs">{{harvnb|Briggs|Cochran|2010|loc=p. 141}}</ref>

:<math>f^{(n)}(x) = \frac{d^ny}{dx^n}.</math>

This notation, for the [[second derivative]], is obtained by using {{math|{{sfrac|''d''|''dx''}}}} 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 {{math|{{sfrac|''dy''|''dx''}}}} as a quotient of [[Differential (mathematics)|differentials]], this should not be done with the higher order forms.<ref>{{harvnb|Swokowski|1983|loc=p. 135}}</ref> However, an alternative Leibniz [[notation for differentiation]] for higher orders allows for this.{{cn|date=March 2024}}

This notation was, however, not used by Leibniz. In print he did not use multi-tiered notation nor numerical exponents (before 1695). To write {{math|''x''<sup>3</sup>}} for instance, he would write {{mvar|xxx}}, 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 {{mvar|dxdx}}. However, Leibniz did use his {{mvar|d}} notation as we would today use operators, namely he would write a second derivative as {{mvar|ddy}} and a third derivative as {{mvar|dddy}}. In 1695 Leibniz started to write {{math|''d''<sup>2</sup>⋅''x''}} and {{math|''d''<sup>3</sup>⋅''x''}} for {{mvar|ddx}} and {{mvar|dddx}} respectively, but [[Guillaume de l'Hôpital|l'Hôpital]], in his textbook on calculus written around the same time, used Leibniz's original forms.<ref>{{harvnb|Cajori|1993|loc=pp. 204-205}}</ref>