Editing Derivative (section)

== Notation ==
{{Main|Notation for differentiation}}
One common way of writing the derivative of a function is [[Leibniz notation]], introduced by [[Gottfried Leibniz|Gottfried Wilhelm Leibniz]] in 1675, which denotes a derivative as the quotient of two [[differential (mathematics)|differentials]], such as <math> dy </math> and {{tmath|1= dx }}.{{sfnm|1a1=Apostol|1y=1967|1p=172 |2a1=Cajori|2y=2007|2p=204}} It is still commonly used when the equation <math>y=f(x)</math> is viewed as a functional relationship between [[dependent and independent variables]]. The first derivative is denoted by {{tmath|1= \textstyle \frac{dy}{dx}  }}, read as "the derivative of <math> y </math> with respect to {{tmath|1= x }}".{{sfn|Moore|Siegel|2013|p=110}} This derivative can alternately be treated as the application of a [[differential operator]] to a function, <math display="inline">\frac{dy}{dx} = \frac{d}{dx} f(x).</math> Higher derivatives are expressed using the notation <math display="inline"> \frac{d^n y}{dx^n} </math> for the <math>n</math>-th derivative of <math>y = f(x)</math>. These are abbreviations for multiple applications of the derivative operator; for example, <math display="inline">\frac{d^2y}{dx^2} = \frac{d}{dx}\Bigl(\frac{d}{dx} f(x)\Bigr).</math>{{sfn|Varberg|Purcell|Rigdon|2007|pp=125–126}} Unlike some alternatives, Leibniz notation involves explicit specification of the variable for differentiation, in the denominator, which removes ambiguity when working with multiple interrelated quantities. The derivative of a [[function composition|composed function]] can be expressed using the [[chain rule]]: if <math>u = g(x)</math> and <math>y = f(g(x))</math> then <math display="inline">\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.</math><ref>In the formulation of calculus in terms of limits, various authors have assigned the <math> du </math> symbol various meanings. Some authors such as {{harvnb|Varberg|Purcell|Rigdon|2007}}, p. 119 and {{harvnb|Stewart|2002}}, p. 177 do not assign a meaning to <math> du </math> by itself, but only as part of the symbol <math display="inline"> \frac{du}{dx} </math>. Others define <math> dx </math> as an independent variable, and define <math> du </math> by {{tmath|1= \textstyle du = dx f'(x) }}.
In [[non-standard analysis]] <math> du </math> is defined as an infinitesimal. It is also interpreted as the [[exterior derivative]] of a function {{tmath|1= u }}. See [[differential (infinitesimal)]] for further information.</ref>

Another common notation for differentiation is by using the [[Prime (symbol)|prime mark]] in the symbol of a function {{tmath|1= f(x) }}. This notation, due to [[Joseph-Louis Lagrange]], is now known as ''prime notation''.{{sfnm|1a1=Schwartzman|1y=1994|1p=[https://books.google.com/books?id=PsH2DwAAQBAJ&pg=PA171 171] |2a1=Cajori|2y=1923|2pp=6–7, 10–12, 21–24}} The first derivative is written as {{tmath|1= f'(x) }}, read as "{{tmath|1= f }} prime of {{tmath|1= x }}, or {{tmath|1= y' }}, read as "{{tmath|1= y }} prime".{{sfnm
 | 1a1 = Moore | 1a2 = Siegel | 1y = 2013 | 1p = 110
 | 2a1 = Goodman | 2y = 1963 | 2pp = 78–79
}} Similarly, the second and the third derivatives can be written as <math> f'' </math> and {{tmath|1= f' ' ' }}, respectively.{{sfnm
 | 1a1 = Varberg | 1a2 = Purcell | 1a3 = Rigdon | 1y = 2007 | 1pp = 125–126
 | 2a1 = Cajori | 2y = 2007 | 2p = 228
}} For denoting the number of higher derivatives beyond this point, some authors use Roman numerals in [[Subscript and superscript|superscript]], whereas others place the number in parentheses, such as <math>f^{\mathrm{iv}}</math> or {{tmath|1= f^{(4)} }}.{{sfnm
 | 1a1 = Choudary | 1a2 = Niculescu | 1y = 2014 | 1p = [https://books.google.com/books?id=I8aPBQAAQBAJ&pg=PA222 222]
 | 2a1 = Apostol | 2y = 1967 | 2p = 171
}} The latter notation generalizes to yield the notation <math>f^{(n)}</math> for the {{tmath|1= n }}th derivative of {{tmath|1= f }}.{{sfn|Varberg|Purcell|Rigdon|2007|pp=125–126}}

In [[Newton's notation]] or the ''dot notation,'' a dot is placed over a symbol to represent a time derivative. If <math> y </math> is a function of {{tmath|1= t }}, then the first and second derivatives can be written as <math>\dot{y}</math> and {{tmath|1= \ddot{y} }}, respectively. This notation is used exclusively for derivatives with respect to time or [[arc length]]. It is typically used in [[differential equation]]s in [[physics]] and [[differential geometry]].{{sfnm
 | 1a1 = Evans | 1y = 1999 | 1p = 63
 | 2a1 = Kreyszig | 2y = 1991 | 2p = 1
}} However, the dot notation becomes unmanageable for high-order derivatives (of order 4 or more) and cannot deal with multiple independent variables.

Another notation is ''D-notation'', which represents the differential operator by the symbol {{tmath|1= D }}.{{sfn|Varberg|Purcell|Rigdon|2007|pp=125–126}} The first derivative is written <math>D f(x)</math> and higher derivatives are written with a superscript, so the <math>n</math>-th derivative is {{tmath|1= D^n f(x)}}. This notation is sometimes called ''Euler notation'', although it seems that [[Leonhard Euler]] did not use it, and the notation was introduced by [[Louis François Antoine Arbogast]].{{sfn|Cajori|1923}} To indicate a partial derivative, the variable differentiated by is indicated with a subscript, for example given the function {{tmath|1= u = f(x, y)}}, its partial derivative with respect to <math>x</math> can be written <math>D_x u</math> or {{tmath|1= D_x f(x,y) }}. Higher partial derivatives can be indicated by superscripts or multiple subscripts, e.g. <math display=inline>D_{xy} f(x,y) = \frac{\partial}{\partial y} \Bigl(\frac{\partial}{\partial x} f(x,y) \Bigr)</math> and {{tmath|1= \textstyle D_{x}^2 f(x,y) = \frac{\partial}{\partial x} \Bigl(\frac{\partial}{\partial x} f(x,y) \Bigr) }}.{{sfnm
 | 1a1 = Apostol | 1y = 1967 | 1p = 172
 | 2a1 = Varberg | 2a2 = Purcell | 2a3 = Rigdon | 2y = 2007 | 2pp = 125–126
}}