Editing Derivative (section)

== Definition ==
=== As a limit ===
A [[function of a real variable]] <math> f(x) </math> is [[Differentiable function|differentiable]] at a point <math> a </math> of its [[domain of a function|domain]], if its domain contains an [[open interval]] containing {{tmath|1= a }}, and the [[limit (mathematics)|limit]]
<math display="block">L=\lim_{h \to 0}\frac{f(a+h)-f(a)}h </math>
exists.{{sfnm|1a1=Apostol |1y=1967 |1p=160 |2a1=Stewart|2y=2002|2p=127 | 3a1=Strang et al.|3y=2023|3p=[https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative 220]}}  This means that, for every positive [[real number]] {{tmath|1= \varepsilon }}, there exists a positive real number <math>\delta</math> such that, for every <math> h </math> such that <math>|h| < \delta</math> and <math>h\ne 0</math> then <math>f(a+h)</math> is defined, and 
<math display="block">\left|L-\frac{f(a+h)-f(a)}h\right|<\varepsilon,</math>
where the vertical bars denote the [[absolute value]]. This is an example of the [[(ε, δ)-definition of limit]].{{sfnm|1a1=Gonick|1y=2012|1p=83|2a1=Thomas et al.|2y=2014|2p=60}}

If the function <math> f </math> is differentiable at {{tmath|1= a }}, that is if the limit <math> L </math> exists, then this limit is called the ''derivative'' of <math> f </math> at <math> a </math>. Multiple notations for the derivative exist.{{sfnm|1a1=Gonick|1y=2012|1p=88 | 2a1=Strang et al.|2y=2023|2p=[https://openstax.org/books/calculus-volume-1/pages/3-2-the-derivative-as-a-function 234]}} The derivative of <math> f </math> at <math> a </math> can be denoted {{tmath|1= f'(a) }}, read as "{{tmath|1= f }} prime of {{tmath|1= a }}"; or it can be denoted {{tmath|1= \textstyle \frac{df}{dx}(a) }}, read as "the derivative of <math> f </math> with respect to <math> x </math> at {{tmath|1= a }}" or "{{tmath|1= df }} by (or over) <math> dx </math> at {{tmath|1= a }}". See ''{{slink||Notation}}'' below. If <math> f </math> is a function that has a derivative at every point in its [[domain of a function|domain]], then a function can be defined by mapping every point <math> x </math> to the value of the derivative of <math> f </math> at <math> x </math>. This function is written <math> f' </math> and is called the ''derivative function'' or the ''derivative of'' {{tmath|1= f }}. The function <math> f </math> sometimes has a derivative at most, but not all, points of its domain. The function whose value at <math> a </math> equals <math> f'(a) </math> whenever <math> f'(a) </math> is defined and elsewhere is undefined is also called the derivative of {{tmath|1= f }}. It is still a function, but its domain may be smaller than the domain of <math> f </math>.{{sfnm
 | 1a1 = Gonick | 1y = 2012 | 1p = 83
 | 2a1 = Strang et al. | 2y = 2023 | 2p = [https://openstax.org/books/calculus-volume-1/pages/3-2-the-derivative-as-a-function 232]
}}

For example, let <math>f</math> be the squaring function: <math>f(x) = x^2</math>. Then the quotient in the definition of the derivative is{{sfn|Gonick|2012|pp=77–80}}
<math display="block">\frac{f(a+h) - f(a)}{h} = \frac{(a+h)^2 - a^2}{h} = \frac{a^2 + 2ah + h^2 - a^2}{h} = 2a + h.</math>
The division in the last step is valid as long as <math>h \neq 0</math>. The closer <math>h</math> is to {{tmath|1= 0 }}, the closer this expression becomes to the value <math>2a</math>. The limit exists, and for every input <math>a</math> the limit is <math>2a</math>. So, the derivative of the squaring function is the doubling function: {{tmath|1= f'(x) = 2x }}.

The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function {{tmath|1= f }}, specifically the points <math>(a,f(a))</math> and <math>(a+h, f(a+h))</math>. As <math>h</math> is made smaller, these points grow closer together, and the slope of this line approaches the limiting value, the slope of the [[tangent]] to the graph of <math>f</math> at <math>a</math>. In other words, the derivative is the slope of the tangent.{{sfnm
 | 1a1 = Thompson | 1y = 1998 | 1pp = 34,104
 | 2a1 = Stewart | 2y = 2002 | 2p = 128
}}

===Using infinitesimals===
One way to think of the derivative <math display="inline">\frac{df}{dx}(a)</math> is as the ratio of an [[infinitesimal]] change in the output of the function <math>f</math> to an infinitesimal change in its input.{{sfn|Thompson|1998|pp=84–85}} In order to make this intuition rigorous, a system of rules for manipulating infinitesimal quantities is required.{{sfn|Keisler|2012|pp=902–904}} The system of [[hyperreal number]]s is a way of treating [[Infinity|infinite]] and infinitesimal quantities.  The hyperreals are an [[Field extension|extension]] of the [[real number]]s that contain numbers greater than anything of the form <math>1 + 1 + \cdots + 1 </math> for any finite number of terms. Such numbers are infinite, and their [[Multiplicative inverse|reciprocal]]s are infinitesimals. The application of hyperreal numbers to the foundations of calculus is called [[nonstandard analysis]]. This provides a way to define the basic concepts of calculus such as the derivative and integral in terms of infinitesimals, thereby giving a precise meaning to the <math>d</math> in the Leibniz notation. Thus, the derivative of <math>f(x)</math> becomes <math display="block">f'(x) = \operatorname{st}\left( \frac{f(x + dx) - f(x)}{dx} \right)</math> for an arbitrary infinitesimal {{tmath|1= dx }}, where <math>\operatorname{st}</math> denotes the [[standard part function]], which "rounds off" each finite hyperreal to the nearest real.{{sfnm
 | 1a1 = Keisler | 1y = 2012 | 1p = 45
 | 2a1 = Henle | 2a2 = Kleinberg | 2y = 2003 | 2p = 66
}} Taking the squaring function <math>f(x) = x^2</math> as an example again,
<math display="block"> \begin{align}
 f'(x) &= \operatorname{st}\left(\frac{x^2 + 2x \cdot dx + (dx)^2 -x^2}{dx}\right) \\
 &= \operatorname{st}\left(\frac{2x \cdot dx +  (dx)^2}{dx}\right) \\
 &= \operatorname{st}\left(\frac{2x \cdot dx}{dx} + \frac{(dx)^2}{dx}\right) \\
 &= \operatorname{st}\left(2x + dx\right) \\
 &= 2x.
\end{align} </math>