Editing Leibniz's notation (section)

===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>