View source for Derivative

{{Short description|Instantaneous rate of change (mathematics)}}
{{distinguish|Derivation (differential algebra)}}
{{other uses|}}
{{pp-semi-indef|small=yes}}
{{good article}}
{{multiple image
| total_width       = 480
| image1            = Tangent to a curve.svg
| caption1          = The [[graph of a function]], drawn in black, and a [[tangent line]] to that graph, drawn in red. The [[slope]] of the tangent line is equal to the derivative of the function at the marked point.
| image2            = Tangent function animation.gif
| caption2          = The derivative at different points of a differentiable function. In this case, the derivative is equal to <math>\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right)</math>.
}}
{{Calculus |differential}}

In [[mathematics]], the '''derivative''' is a fundamental tool that quantifies the sensitivity to change of a [[Function (mathematics)|function]]'s output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the [[slope]] of the [[Tangent|tangent line]] to the [[graph of a function|graph of the function]] at that point. The tangent line is the best [[linear approximation]] of the function near that input value. For this reason, the derivative is often described as the '''instantaneous rate of change''', the ratio of the instantaneous change in the dependent variable to that of the independent variable.{{sfnm|1a1=Apostol|1y=1967|1p=160 |2a1=Stewart|2y=2002|2pp=129–130 |3a1=Strang et al.|3y=2023|3p=[https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative 224]}} The process of finding a derivative is called '''differentiation'''.

There are multiple different [[mathematical notation|notations]] for differentiation. ''[[Leibniz notation]]'', named after [[Gottfried Wilhelm Leibniz]], is represented as the ratio of two [[Differential (mathematics)|differentials]], whereas ''prime notation'' is written by adding a [[prime mark]]. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. The [[higher order derivative]]s can be applied in physics; for example, while the first derivative of the position of a moving object with respect to [[time]] is the object's [[velocity]], how the position changes as time advances, the second derivative is the object's [[acceleration]], how the velocity changes as time advances.

Derivatives can be generalized to [[function of several real variables|functions of several real variables]]. In this case, the derivative is reinterpreted as a [[linear transformation]] whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The [[Jacobian matrix]] is the [[matrix (mathematics)|matrix]] that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables.  It can be calculated in terms of the [[partial derivative]]s with respect to the independent variables.  For a [[real-valued function]] of several variables, the Jacobian matrix reduces to the [[gradient vector]].

== 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>

== Continuity and differentiability ==
{{multiple image
| total_width       = 480
| image1            = Right-continuous.svg
| caption1          = This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a [[jump discontinuity]]).
| image2            = Absolute value.svg
| caption2          = The absolute value function is continuous but fails to be differentiable at {{math|''x'' {{=}} 0}} since the tangent slopes do not approach the same value from the left as they do from the right.
}}

If <math> f </math> is [[differentiable]] at {{tmath|1= a }}, then <math> f </math> must also be [[continuous function|continuous]] at <math> a </math>.{{sfnm|1a1=Gonick|1y=2012|1p=156 |2a1=Thomas et al.|2y=2014|2p=114 |3a1=Strang et al. |3y=2023 |3p=[https://openstax.org/books/calculus-volume-1/pages/3-2-the-derivative-as-a-function 237] }} As an example, choose a point <math> a </math> and let <math> f </math> be the [[step function]] that returns the value 1 for all <math> x </math> less than {{tmath|1= a }}, and returns a different value 10 for all <math> x </math> greater than or equal to <math> a </math>.  The function <math> f </math> cannot have a derivative at <math> a </math>. If <math> h </math> is negative, then <math> a + h </math> is on the low part of the step, so the secant line from <math> a </math> to <math> a + h </math> is very steep; as <math> h </math> tends to zero, the slope tends to infinity. If <math> h </math> is positive, then <math> a + h </math> is on the high part of the step, so the secant line from <math> a </math> to <math> a + h </math> has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. However, even if a function is continuous at a point, it may not be differentiable there. For example, the [[absolute value]] function given by <math> f(x) = |x| </math> is continuous at {{tmath|1= x = 0 }}, but it is not differentiable there. If <math> h </math> is positive, then the slope of the secant line from 0 to <math> h </math> is one; if <math> h </math> is negative, then the slope of the secant line from <math> 0 </math> to <math> h </math> is {{tmath|1= -1 }}.{{sfnm|1a1=Gonick|1y=2012|1p=149 |2a1=Thomas et al.|2y=2014|2p=113 |3a1=Strang et al. |3y=2023 |3p=[https://openstax.org/books/calculus-volume-1/pages/3-2-the-derivative-as-a-function 237]}}  This can be seen graphically as a "kink" or a "cusp" in the graph at <math>x=0</math>. Even a function with a smooth graph is not differentiable at a point where its [[Vertical tangent|tangent is vertical]]: For instance, the function given by <math> f(x) = x^{1/3} </math> is not differentiable at <math> x = 0 </math>. In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.{{sfnm|1a1=Gonick|1y=2012|1p=156 |2a1=Thomas et al.|2y=2014|2p=114 |3a1=Strang et al. |3y=2023 |3pp=[https://openstax.org/books/calculus-volume-1/pages/3-2-the-derivative-as-a-function 237–238]}}

Most functions that occur in practice have derivatives at all points or [[Almost everywhere|almost every]] point. Early in the [[history of calculus]], many mathematicians assumed that a continuous function was differentiable at most points.{{sfnm
 | 1a1 = Jašek | 1y = 1922
 | 2a1 = Jarník | 2y = 1922
 | 3a1 = Rychlík | 3y = 1923
}} Under mild conditions (for example, if the function is a [[monotone function|monotone]] or a [[Lipschitz function]]), this is true. However, in 1872, Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere.  This example is now known as the [[Weierstrass function]].{{sfn|David|2018}} In 1931, [[Stefan Banach]] proved that the set of functions that have a derivative at some point is a [[meager set]] in the space of all continuous functions. Informally, this means that hardly any random continuous functions have a derivative at even one point.<ref>{{harvnb|Banach|1931}}, cited in {{harvnb|Hewitt|Stromberg|1965}}.</ref>

== 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
}}

== Rules of computation ==
{{Main|Differentiation rules}}
In principle, the derivative of a function can be computed from the definition by considering the difference quotient and computing its limit. Once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using ''rules'' for obtaining derivatives of more complicated functions from simpler ones. This process of finding a derivative is known as '''differentiation'''.{{sfn|Apostol|1967|p=160}}

=== Rules for basic functions ===
The following are the rules for the derivatives of the most common basic functions. Here, <math> a </math> is a real number, and <math> e </math> is [[e (mathematical constant)|the base of the natural logarithm, approximately {{nowrap|1=2.71828}}]].<ref>{{harvnb|Varberg|Purcell|Rigdon|2007}}. See p. 133 for the power rule, pp. 115–116 for the trigonometric functions, p. 326 for the natural logarithm, pp. 338–339 for exponential with base {{tmath|1= e }}, p. 343 for the exponential with base {{tmath|1= a }}, p. 344 for the logarithm with base {{tmath|1= a }}, and p. 369 for the inverse of trigonometric functions.</ref> 
* ''[[Power rule|Derivatives of powers]]'':
*: <math> \frac{d}{dx}x^a = ax^{a-1} </math>
<!--DO NOT ADD TO THIS LIST-->
* ''Functions of [[Exponential function|exponential]], [[natural logarithm]], and [[logarithm]] with general base'':
*: <math> \frac{d}{dx}e^x = e^x </math>
*: <math> \frac{d}{dx}a^x = a^x\ln(a)  </math>, for <math> a > 0 </math>
*: <math> \frac{d}{dx}\ln(x) = \frac{1}{x} </math>, for <math> x > 0 </math>
*: <math> \frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}  </math>, for <math> x, a > 0 </math>
<!--DO NOT ADD TO THIS LIST-->
* ''[[Trigonometric functions]]'':
*: <math> \frac{d}{dx}\sin(x) = \cos(x) </math>
*: <math> \frac{d}{dx}\cos(x) = -\sin(x) </math>
*: <math> \frac{d}{dx}\tan(x) = \sec^2(x) = \frac{1}{\cos^2(x)} = 1 + \tan^2(x) </math>
<!--DO NOT ADD TO THIS LIST-->
* ''[[Inverse trigonometric functions]]'':
*: <math> \frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}} </math>, for <math> -1 < x < 1 </math>
*: <math> \frac{d}{dx}\arccos(x)= -\frac{1}{\sqrt{1-x^2}} </math>, for <math> -1 < x < 1 </math>
*: <math> \frac{d}{dx}\arctan(x)= \frac{1}{{1+x^2}} </math>
<!--DO NOT ADD TO THIS LIST-->

=== Rules for combined functions <span class="anchor" id="Rules"></span> ===
Given that the <math> f </math> and <math> g </math> are the functions. The following are some of the most basic rules for deducing the derivative of functions from derivatives of basic functions.<ref> For constant rule and sum rule, see {{harvnb|Apostol|1967|pp=161, 164}}, respectively. For the product rule, quotient rule, and chain rule, see {{harvnb|Varberg|Purcell|Rigdon|2007|pp=111–112, 119}}, respectively. For the special case of the product rule, that is, the product of a constant and a function, see {{harvnb|Varberg|Purcell|Rigdon|2007|pp=108–109}}.</ref>
* ''Constant rule'': if <math>f</math> is constant, then for all {{tmath|1= x }},
*: <math>f'(x) = 0. </math>
* ''[[Linearity of differentiation|Sum rule]]'':
*: <math>(\alpha f + \beta g)' = \alpha f' + \beta g' </math> for all functions <math>f</math> and <math>g</math> and all real numbers <math>\alpha</math> and {{tmath|1= \beta}}.
* ''[[Product rule]]'':
*: <math>(fg)' = f 'g + fg' </math> for all functions <math>f</math> and {{tmath|1= g }}. As a special case, this rule includes the fact <math>(\alpha f)' = \alpha f'</math> whenever <math>\alpha</math> is a constant because <math>\alpha' f = 0 \cdot f = 0</math> by the constant rule.
* ''[[Quotient rule]]'':
*: <math>\left(\frac{f}{g} \right)' = \frac{f'g - fg'}{g^2}</math>  for all functions <math>f</math> and <math>g</math> at all inputs where {{nowrap|''g'' ≠ 0}}.
* ''[[Chain rule]]'' for [[Function composition|composite functions]]: If {{tmath|1= f(x) = h(g(x)) }}, then
*: <math>f'(x) = h'(g(x)) \cdot g'(x). </math>

=== Computation example ===
The derivative of the function given by <math>f(x) = x^4 + \sin \left(x^2\right) - \ln(x) e^x + 7</math> is
<math display="block"> \begin{align}
 f'(x) &= 4 x^{(4-1)}+ \frac{d\left(x^2\right)}{dx}\cos \left(x^2\right) - \frac{d\left(\ln {x}\right)}{dx} e^x - \ln(x) \frac{d\left(e^x\right)}{dx} + 0 \\
 &= 4x^3 + 2x\cos \left(x^2\right) - \frac{1}{x} e^x - \ln(x) e^x.
\end{align} </math>
Here the second term was computed using the [[chain rule]] and the third term using the [[product rule]]. The known derivatives of the elementary functions <math> x^2 </math>, <math> x^4 </math>, <math> \sin (x) </math>, <math> \ln (x) </math>, and <math> \exp(x) = e^x </math>, as well as the constant <math> 7 </math>, were also used.

== Higher-order derivatives <span class="anchor" id="order of derivation"></span><span class="anchor" id="Order"></span> ==
''Higher order derivatives'' are the result of differentiating a function repeatedly. Given that <math> f </math> is a differentiable function, the derivative of <math> f </math> is the first derivative, denoted as {{tmath|1= f' }}. The derivative of <math> f' </math> is the [[second derivative]], denoted as {{tmath|1= f' ' }}, and the derivative of <math> f'' </math> is the [[third derivative]], denoted as {{tmath|1= f' ' ' }}. By continuing this process, if it exists, the {{tmath|1= n }}th derivative is the derivative of the {{tmath|1= (n - 1) }}th derivative or the ''derivative of order {{tmath|1= n }}''. As has been [[#Notation|discussed above]], the generalization of derivative of a function <math> f </math> may be denoted as {{tmath|1= f^{(n)} }}.{{sfnm
 | 1a1 = Apostol | 1y = 1967 | 1p = 160
 | 2a1 = Varberg | 2a2 = Purcell | 2a3 = Rigdon | 2y = 2007 | 2pp = 125–126
}}  A function that has <math> k </math> successive derivatives is called ''<math> k </math> times differentiable''. If the {{nowrap|1=<math> k </math>-}}th derivative is continuous, then the function is said to be of [[differentiability class]] {{tmath|1= C^k }}.{{sfn|Warner|1983|p=5}} A function that has infinitely many derivatives is called ''infinitely differentiable'' or ''[[smoothness|smooth]]''.{{sfn|Debnath|Shah|2015|p=[https://books.google.com/books?id=qPuWBQAAQBAJ&pg=PA40 40]}} Any [[polynomial]] function is infinitely differentiable; taking derivatives repeatedly will eventually result in a [[constant function]], and all subsequent derivatives of that function are zero.{{sfn|Carothers|2000|p=[https://books.google.com/books?id=4VFDVy1NFiAC&pg=PA176 176]}}

One [[Differential calculus#Applications of derivatives|application of higher-order derivatives]] is in [[physics]]. Suppose that a function represents the position of an object at the time. The first derivative of that function is the [[velocity]] of an object with respect to time, the second derivative of the function is the [[acceleration]] of an object with respect to time,{{sfn|Apostol|1967|p=160}} and the third derivative is the [[jerk (physics)|jerk]].{{sfn|Stewart|2002|p=193}}

== In other dimensions ==
{{See also|Vector calculus|Multivariable calculus}}

=== Vector-valued functions ===
A [[vector-valued function]] <math> \mathbf{y} </math> of a real variable sends real numbers to vectors in some [[vector space]] <math> \R^n </math>. A vector-valued function can be split up into its coordinate functions <math> y_1(t), y_2(t), \dots, y_n(t) </math>, meaning that <math> \mathbf{y} = (y_1(t), y_2(t), \dots, y_n(t))</math>. This includes, for example, [[parametric curve]]s in <math> \R^2 </math> or <math> \R^3 </math>. The coordinate functions are real-valued functions, so the above definition of derivative applies to them. The derivative of <math> \mathbf{y}(t) </math> is defined to be the [[Vector (geometric)|vector]], called the [[Differential geometry of curves|tangent vector]], whose coordinates are the derivatives of the coordinate functions. That is,{{sfn|Stewart|2002|p=893}}
<math display="block"> \mathbf{y}'(t)=\lim_{h\to 0}\frac{\mathbf{y}(t+h) - \mathbf{y}(t)}{h}, </math>
if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of <math> \mathbf{y} </math> exists for every value of {{tmath|1= t }}, then <math> \mathbf{y}' </math> is another vector-valued function.{{sfn|Stewart|2002|p=893}}

=== Partial derivatives ===
{{Main|Partial derivative}}
Functions can depend upon [[function (mathematics)#Multivariate function|more than one variable]]. A [[partial derivative]] of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in [[vector calculus]] and [[differential geometry]]. As with ordinary derivatives, multiple notations exist: the partial derivative of a function <math>f(x, y, \dots)</math> with respect to the variable <math>x</math> is variously denoted by
{{block indent | em = 1.2 | text = <math>f_x</math>, <math>f'_x</math>, <math>\partial_x f</math>, <math>\frac{\partial}{\partial x}f</math>, or <math>\frac{\partial f}{\partial x}</math>,}}
among other possibilities.{{sfnm
 | 1a1 = Stewart | 1y = 2002 | 1p = [https://archive.org/details/calculus0000stew/page/947/mode/1up 947]
 | 2a1 = Christopher | 2y = 2013 | 2p = 682
}} It can be thought of as the rate of change of the function in the <math>x</math>-direction.{{sfn|Stewart|2002|p=[https://archive.org/details/calculus0000stew/page/949 949]}} Here [[∂]] is a rounded ''d'' called the '''partial derivative symbol'''.  To distinguish it from the letter ''d'', ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee".{{sfnm
 | 1a1 = Silverman | 1y = 1989 | 1p = [https://books.google.com/books?id=CQ-kqE9Yo9YC&pg=PA216 216]
 | 2a1 = Bhardwaj | 2y= 2005 | 2loc = See [https://books.google.com/books?id=qSlGMwpNueoC&pg=SA6-PA4 p. 6.4]
}} For example, let {{tmath|1= f(x,y) = x^2 + xy + y^2 }}, then the partial derivative of function <math> f </math> with respect to both variables <math> x </math> and <math> y </math> are, respectively:
<math display="block"> \frac{\partial f}{\partial x} = 2x + y, \qquad \frac{\partial f}{\partial y} = x + 2y.</math>
In general, the partial derivative of a function <math> f(x_1, \dots, x_n) </math> in the direction <math> x_i </math> at the point <math>(a_1, \dots, a_n) </math> is defined to be:{{sfn|Mathai|Haubold|2017|p=[https://books.google.com/books?id=v20uDwAAQBAJ&pg=PA52 52]}}
<math display="block">\frac{\partial f}{\partial x_i}(a_1,\ldots,a_n) = \lim_{h \to 0}\frac{f(a_1,\ldots,a_i+h,\ldots,a_n) - f(a_1,\ldots,a_i,\ldots,a_n)}{h}.</math>

This is fundamental for the study of the [[functions of several real variables]]. Let <math> f(x_1, \dots, x_n) </math> be such a [[real-valued function]]. If all partial derivatives <math> f </math> with respect to <math> x_j </math> are defined at the point {{tmath|1= (a_1, \dots, a_n) }}, these partial derivatives define the vector
<math display="block">\nabla f(a_1, \ldots, a_n) = \left(\frac{\partial f}{\partial x_1}(a_1, \ldots, a_n), \ldots, \frac{\partial f}{\partial x_n}(a_1, \ldots, a_n)\right),</math>
which is called the [[gradient]] of <math> f </math> at <math> a </math>. If <math> f </math> is differentiable at every point in some domain, then the gradient is a [[vector-valued function]] <math> \nabla f </math> that maps the point <math> (a_1, \dots, a_n) </math> to the vector <math> \nabla f(a_1, \dots, a_n) </math>. Consequently, the gradient determines a [[vector field]].{{sfn|Gbur|2011|pp=36–37}}

=== Directional derivatives ===
{{Main|Directional derivative}}

If <math> f </math> is a real-valued function on {{tmath|1= \R^n }}, then the partial derivatives of <math> f </math> measure its variation in the direction of the coordinate axes. For example, if <math> f </math> is a function of <math> x </math> and {{tmath|1= y }}, then its partial derivatives measure the variation in <math> f </math> in the <math> x </math> and <math> y </math> direction. However, they do not directly measure the variation of <math> f </math> in any other direction, such as along the diagonal line {{tmath|1= y = x }}. These are measured using directional derivatives. Given a vector {{tmath|1= \mathbf{v} = (v_1,\ldots,v_n) }}, then the [[directional derivative]] of <math> f </math> in the direction of <math> \mathbf{v} </math> at the point <math> \mathbf{x} </math> is:{{sfn|Varberg|Purcell|Rigdon|2007|p=642}}
<math display="block"> D_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \rightarrow 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h}}.</math>
<!--In some cases, it may be easier to compute or estimate the directional derivative after changing the length of the vector.  Often this is done to turn the problem into the computation of a directional derivative in the direction of a unit vector.  To see how this works, suppose that {{nowrap|1='''v''' = ''λ'''''u'''}} where '''u''' is a unit vector in the direction of '''v'''.  Substitute {{nowrap|1=''h'' = ''k''/''λ''}} into the difference quotient.  The difference quotient becomes:
:<math>\frac{f(\mathbf{x} + (k/\lambda)(\lambda\mathbf{u})) - f(\mathbf{x})}{k/\lambda}
= \lambda\cdot\frac{f(\mathbf{x} + k\mathbf{u}) - f(\mathbf{x})}{k}.</math>
This is ''λ'' times the difference quotient for the directional derivative of ''f'' with respect to '''u'''. Furthermore, taking the limit as ''h'' tends to zero is the same as taking the limit as ''k'' tends to zero because ''h'' and ''k'' are multiples of each other.  Therefore, {{nowrap|1=''D''<sub>'''v'''</sub>(''f'') = λ''D''<sub>'''u'''</sub>(''f'')}}.  Because of this rescaling property, directional derivatives are frequently considered only for unit vectors.-->

If all the partial derivatives of <math> f </math> exist and are continuous at {{tmath|1= \mathbf{x} }}, then they determine the directional derivative of <math> f </math> in the direction <math> \mathbf{v} </math> by the formula:{{sfn|Guzman|2003|p=[https://books.google.com/books?id=aI_qBwAAQBAJ&pg=PA35 35]}}
<math display="block"> D_{\mathbf{v}}{f}(\mathbf{x}) = \sum_{j=1}^n v_j \frac{\partial f}{\partial x_j}. </math>

=== Total derivative and Jacobian matrix ===
{{Main|Total derivative}}

When <math> f </math> is a function from an open subset of <math> \R^n </math> to {{tmath|1= \R^m }}, then the directional derivative of <math> f </math> in a chosen direction is the best linear approximation to <math> f </math> at that point and in that direction. However, when {{tmath|1= n > 1 }}, no single directional derivative can give a complete picture of the behavior of <math> f </math>. The total derivative gives a complete picture by considering all directions at once. That is, for any vector <math> \mathbf{v} </math> starting at {{tmath|1= \mathbf{a} }}, the linear approximation formula holds:{{sfn|Davvaz|2023|p=[https://books.google.com/books?id=ofzKEAAAQBAJ&pg=PA266 266]}}
<math display="block">f(\mathbf{a} + \mathbf{v}) \approx f(\mathbf{a}) + f'(\mathbf{a})\mathbf{v}.</math>
Similarly with the single-variable derivative, <math> f'(\mathbf{a}) </math> is chosen so that the error in this approximation is as small as possible. The total derivative of <math> f </math> at <math> \mathbf{a} </math> is the unique linear transformation <math> f'(\mathbf{a}) \colon \R^n \to \R^m </math> such that{{sfn|Davvaz|2023|p=[https://books.google.com/books?id=ofzKEAAAQBAJ&pg=PA266 266]}}
<math display="block">\lim_{\mathbf{h}\to 0} \frac{\lVert f(\mathbf{a} + \mathbf{h}) - (f(\mathbf{a}) + f'(\mathbf{a})\mathbf{h})\rVert}{\lVert\mathbf{h}\rVert} = 0.</math>
Here <math> \mathbf{h} </math> is a vector in {{tmath|1= \R^n }}, so the norm in the denominator is the standard length on <math> \R^n </math>. However, <math> f'(\mathbf{a}) \mathbf{h} </math> is a vector in {{tmath|1= \R^m }}, and the norm in the numerator is the standard length on <math> \R^m </math>.{{sfn|Davvaz|2023|p=[https://books.google.com/books?id=ofzKEAAAQBAJ&pg=PA266 266]}} If <math> v </math> is a vector starting at {{tmath|1= a }}, then <math> f'(\mathbf{a}) \mathbf{v} </math> is called the [[pushforward (differential)|pushforward]] of <math> \mathbf{v} </math> by <math> f </math>.{{sfn|Lee|2013|p=72}}

If the total derivative exists at {{tmath|1= \mathbf{a} }}, then all the partial derivatives and directional derivatives of <math> f </math> exist at {{tmath|1= \mathbf{a} }}, and for all {{tmath|1= \mathbf{v} }}, <math> f'(\mathbf{a})\mathbf{v} </math> is the directional derivative of <math> f </math> in the direction {{tmath|1= \mathbf{v} }}.  If <math> f </math> is written using coordinate functions, so that {{tmath|1= f = (f_1, f_2, \dots, f_m) }}, then the total derivative can be expressed using the partial derivatives as a [[matrix (mathematics)|matrix]]. This matrix is called the [[Jacobian matrix]] of <math> f </math> at <math> \mathbf{a} </math>:{{sfn|Davvaz|2023|p=[https://books.google.com/books?id=ofzKEAAAQBAJ&pg=PA267 267]}}
<math display="block">f'(\mathbf{a}) = \operatorname{Jac}_{\mathbf{a}} = \left(\frac{\partial f_i}{\partial x_j}\right)_{ij}.</math>
<!-- The existence of the total derivative <math> f(\mathbf{a}) </math> is strictly stronger than the existence of all the partial derivatives, but if the partial derivatives exist and are continuous, then the total derivative exists, is given by the Jacobian, and depends continuously on {{tmath|1= \mathbf{a} }}.

The definition of the total derivative subsumes the definition of the derivative in one variable. That is, if <math> f </math> is a real-valued function of a real variable, then the total derivative exists if and only if the usual derivative exists. The Jacobian matrix reduces to a {{nowrap|1=1×1}} matrix whose only entry is the derivative {{tmath|1= f'(x) }}. This {{nowrap|1=1×1}} matrix satisfies the property that
<math display="block"> f(a+h) \approx f(a) + f'(a)h.</math>
Up to changing variables, this is the statement that the function <math>x \mapsto f(a) + f'(a)(x-a)</math> is the best linear approximation to <math> f </math> at {{tmath|1= a }}.{{cn|date=January 2024}}

The total derivative of a function does not give another function in the same way as the one-variable case. This is because the total derivative of a multivariable function has to record much more information than the derivative of a single-variable function. Instead, the total derivative gives a function from the [[tangent bundle]] of the source to the tangent bundle of the target.{{cn|date=January 2024}}

The natural analog of second, third, and higher-order total derivatives is not a linear transformation, is not a function on the tangent bundle, and is not built by repeatedly taking the total derivative. The analog of a higher-order derivative, called a [[jet (mathematics)|jet]], cannot be a linear transformation because higher-order derivatives reflect subtle geometric information, such as concavity, which cannot be described in terms of linear data such as vectors.  It cannot be a function on the tangent bundle because the tangent bundle only has room for the base space and the directional derivatives. Because jets capture higher-order information, they take as arguments additional coordinates representing higher-order changes in direction. The space determined by these additional coordinates is called the [[jet bundle]].  The relation between the total derivative and the partial derivatives of a function is paralleled in the relation between the {{nowrap|1=<math> k </math>-}}th order jet of a function and its partial derivatives of order less than or equal to {{tmath|1= k }}.{{cn|date=January 2024}}

By repeatedly taking the total derivative, one obtains higher versions of the [[Fréchet derivative]], specialized to {{tmath|1= \R^p }}. The {{nowrap|1=<math> k </math>-}}th order total derivative may be interpreted as a map
<math display="block"> D^k f: \mathbb{R}^n \to L^k(\mathbb{R}^n \times \cdots \times \mathbb{R}^n, \mathbb{R}^m), </math>
which takes a point <math> \mathbf{x} </math> in <math> \R^n </math> and assigns to it an element of the space of {{nowrap|1=<math> k </math>-}}linear maps from <math> \R^n </math> to <math> \R^m </math> &mdash; the "best" (in a certain precise sense) {{nowrap|1=<math> k </math>-}}linear approximation to <math> f </math> at that point. By precomposing it with the [[Diagonal functor|diagonal map]] <math> \Delta </math>, <math> \mathbf{x} \to (\mathbf{x}, \mathbf{x}) </math>, a generalized Taylor series may be begun as
<math display="block">\begin{align}
 f(\mathbf{x}) & \approx f(\mathbf{a}) + (D f)(\mathbf{x-a}) + \left(D^2 f\right)(\Delta(\mathbf{x-a})) + \cdots\\
 & = f(\mathbf{a}) + (D f)(\mathbf{x - a}) + \left(D^2 f\right)(\mathbf{x - a}, \mathbf{x - a})+ \cdots\\
 & = f(\mathbf{a}) + \sum_i (D f)_i (x_i-a_i) + \sum_{j, k} \left(D^2 f\right)_{j k} (x_j-a_j) (x_k-a_k) + \cdots
\end{align}</math>
where <math> f(\mathbf{a}) </math> is identified with a constant function, <math> x_i - a_i </math> are the components of the vector <math> \mathbf{x}- \mathbf{a} </math>, and <math> (Df)_i </math> and <math> (D^2 f)_{jk} </math> are the components of <math> Df </math> and <math> D^2 f </math> as linear transformations.{{cn|date=January 2024}}-->

== Generalizations ==
{{Main|Generalizations of the derivative}}

The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a [[linear approximation]] of the function at that point.
* An important generalization of the derivative concerns [[complex function]]s of [[Complex number|complex variable]]s, such as functions from (a domain in) the complex numbers <math>\C</math> to {{tmath|1= \C }}. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition.{{sfn|Roussos|2014|p=303}} If <math>\C</math> is identified with <math>\R^2</math> by writing a complex number <math>z</math> as {{tmath|1= x+iy }} then a differentiable function from <math>\C</math> to <math>\C</math> is certainly differentiable as a function from <math>\R^2</math> to <math>\R^2</math> (in the sense that its partial derivatives all exist), but the converse is not true in general: the complex derivative only exists if the real derivative is ''complex linear'' and this imposes relations between the partial derivatives called the [[Cauchy–Riemann equations]] – see [[holomorphic function]]s.{{sfn|Gbur|2011|pp=261–264}}
* Another generalization concerns functions between [[smooth manifold|differentiable or smooth manifolds]]. Intuitively speaking such a manifold <math>M</math> is a space that can be approximated near each point <math>x</math> by a vector space called its [[tangent space]]: the prototypical example is a [[smooth surface]] in {{tmath|1= \R^3 }}. The derivative (or differential) of a (differentiable) map <math>f:M\to N</math> between manifolds, at a point <math>x</math> in {{tmath|1= M }}, is then a [[linear map]] from the tangent space of <math>M</math> at <math>x</math> to the tangent space of <math>N</math> at {{tmath|1= f(x) }}. The derivative function becomes a map between the [[tangent bundle]]s of <math>M</math> and {{tmath|1= N }}. This definition is used in [[differential geometry]].{{sfn|Gray|Abbena|Salamon|2006|p=[https://books.google.com/books?id=owEj9TMYo7IC&pg=PA826 826]}}
* Differentiation can also be defined for maps between [[vector space]], such as [[Banach space]], in which those generalizations are the [[Gateaux derivative]] and the [[Fréchet derivative]].<ref>{{harvnb|Azegami|2020}}. See p. [https://books.google.com/books?id=e08AEAAAQBAJ&pg=PA209 209] for the Gateaux derivative, and p. [https://books.google.com/books?id=e08AEAAAQBAJ&pg=PA211 211] for the Fréchet derivative.</ref>
* One deficiency of the classical derivative is that very many functions are not differentiable. Nevertheless, there is a way of extending the notion of the derivative so that all [[continuous function|continuous]] functions and many other functions can be differentiated using a concept known as the [[weak derivative]]. The idea is to embed the continuous functions in a larger space called the space of [[distribution (mathematics)|distributions]] and only require that a function is differentiable "on average".{{sfn|Funaro|1992|pp=[https://books.google.com/books?id=CX4SXf3mdeUC&pg=PA84 84–85]}}
* Properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology; an example is [[differential algebra]]. Here, it consists of the derivation of some topics in abstract algebra, such as [[Ring (mathematics)|rings]], [[Ideal (ring theory)|ideals]], [[Field (mathematics)|field]], and so on.{{sfn|Kolchin|1973|pp=[https://books.google.com/books?id=yDCfhIjka-8C&pg=PA58 58], [https://books.google.com/books?id=yDCfhIjka-8C&pg=PA126 126]}}
* The discrete equivalent of differentiation is [[finite difference]]s. The study of differential calculus is unified with the calculus of finite differences in [[time scale calculus]].{{sfn|Georgiev|2018|p=[https://books.google.com/books?id=OJJVDwAAQBAJ&pg=PA8 8]}}
* The [[arithmetic derivative]] involves the function that is defined for the [[Integer|integers]] by the [[prime factorization]]. This is an analogy with the product rule.{{sfn|Barbeau|1961}}

== See also ==
* [[Covariant derivative]]
* [[Derivation (differential algebra)|Derivation]]
* [[Exterior derivative]]
* [[Functional derivative]]
* [[Integral]]
* [[Lie derivative ]]

== Notes ==
{{reflist}}

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{{refend}}

== External links ==
{{Sister project links|Differentiation|auto=1|wikt=y|b=y|v=y}}
* {{springer|title=Derivative|id=p/d031260}}
* [[Khan Academy]]: [https://www.khanacademy.org/math/differential-calculus/taking-derivatives/intro_differential_calc/v/newton-leibniz-and-usain-bolt "Newton, Leibniz, and Usain Bolt"]
* {{MathWorld |title=Derivative |id=Derivative}}
* [http://www.wolframalpha.com/calculators/derivative-calculator/ Online Derivative Calculator] from [[Wolfram Alpha]].

{{Calculus topics}}
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[[Category:Mathematical analysis]]
[[Category:Differential calculus]]
[[Category:Functions and mappings]]
[[Category:Linear operators in calculus]]
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