Derivative

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In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a 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 line to the 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.Template:Sfnm The process of finding a derivative is called differentiation.

There are multiple different notations for differentiation. Leibniz notation, named after Gottfried Wilhelm Leibniz, is represented as the ratio of two 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 derivatives 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 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 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 derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

DefinitionEdit

As a limitEdit

A function of a real variable <math> f(x) </math> is differentiable at a point <math> a </math> of its domain, if its domain contains an open interval containing Template:Tmath, and the limit <math display="block">L=\lim_{h \to 0}\frac{f(a+h)-f(a)}h </math> exists.Template:Sfnm This means that, for every positive real number Template:Tmath, 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.Template:Sfnm

If the function <math> f </math> is differentiable at Template:Tmath, 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.Template:Sfnm The derivative of <math> f </math> at <math> a </math> can be denoted Template:Tmath, read as "Template:Tmath prime of Template:Tmath"; or it can be denoted Template:Tmath, read as "the derivative of <math> f </math> with respect to <math> x </math> at Template:Tmath" or "Template:Tmath by (or over) <math> dx </math> at Template:Tmath". See Template:Slink below. If <math> f </math> is a function that has a derivative at every point in its 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 Template:Tmath. 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 Template:Tmath. It is still a function, but its domain may be smaller than the domain of <math> f </math>.Template:Sfnm

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 isTemplate:Sfn <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 Template:Tmath, 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: Template:Tmath.

The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function Template:Tmath, 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.Template:Sfnm

Using infinitesimalsEdit

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.Template:Sfn In order to make this intuition rigorous, a system of rules for manipulating infinitesimal quantities is required.Template:Sfn The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals are an extension of the real numbers 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 reciprocals 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 Template:Tmath, where <math>\operatorname{st}</math> denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real.Template:Sfnm 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 differentiabilityEdit

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If <math> f </math> is differentiable at Template:Tmath, then <math> f </math> must also be continuous at <math> a </math>.Template:Sfnm 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 Template:Tmath, 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 Template:Tmath, 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 Template:Tmath.Template:Sfnm 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 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.Template:Sfnm

Most functions that occur in practice have derivatives at all points or almost every point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points.Template:Sfnm Under mild conditions (for example, if the function is a 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.Template:Sfn 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>Template:Harvnb, cited in Template:Harvnb.</ref>

NotationEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} One common way of writing the derivative of a function is Leibniz notation, introduced by Gottfried Wilhelm Leibniz in 1675, which denotes a derivative as the quotient of two differentials, such as <math> dy </math> and Template:Tmath.Template:Sfnm 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 Template:Tmath, read as "the derivative of <math> y </math> with respect to Template:Tmath".Template:Sfn 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>Template:Sfn 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 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 Template:Harvnb, p. 119 and Template:Harvnb, 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 Template:Tmath. In non-standard analysis <math> du </math> is defined as an infinitesimal. It is also interpreted as the exterior derivative of a function Template:Tmath. See differential (infinitesimal) for further information.</ref>

Another common notation for differentiation is by using the prime mark in the symbol of a function Template:Tmath. This notation, due to Joseph-Louis Lagrange, is now known as prime notation.Template:Sfnm The first derivative is written as Template:Tmath, read as "Template:Tmath prime of Template:Tmath", or Template:Tmath, read as "Template:Tmath prime".Template:Sfnm Similarly, the second and the third derivatives can be written as <math> f </math> and Template:Tmath, respectively.Template:Sfnm For denoting the number of higher derivatives beyond this point, some authors use Roman numerals in superscript, whereas others place the number in parentheses, such as <math>f^{\mathrm{iv}}</math> or Template:Tmath.Template:Sfnm The latter notation generalizes to yield the notation <math>f^{(n)}</math> for the Template:Tmathth derivative of Template:Tmath.Template:Sfn

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 Template:Tmath, then the first and second derivatives can be written as <math>\dot{y}</math> and Template:Tmath, respectively. This notation is used exclusively for derivatives with respect to time or arc length. It is typically used in differential equations in physics and differential geometry.Template:Sfnm 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 Template:Tmath.Template:Sfn 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 Template:Tmath. 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.Template:Sfn To indicate a partial derivative, the variable differentiated by is indicated with a subscript, for example given the function Template:Tmath, its partial derivative with respect to <math>x</math> can be written <math>D_x u</math> or Template:Tmath. 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 Template:Tmath.Template:Sfnm

Rules of computationEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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.Template:Sfn

Rules for basic functionsEdit

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 Template:Nowrap]].<ref>Template:Harvnb. 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 Template:Tmath, p. 343 for the exponential with base Template:Tmath, p. 344 for the logarithm with base Template:Tmath, and p. 369 for the inverse of trigonometric functions.</ref>

Derivatives of powers:
<math> \frac{d}{dx}x^a = ax^{a-1} </math>
Functions of 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>
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>
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}Template:1+x^2 </math>

Rules for combined functions Edit

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 Template:Harvnb, respectively. For the product rule, quotient rule, and chain rule, see Template:Harvnb, respectively. For the special case of the product rule, that is, the product of a constant and a function, see Template:Harvnb.</ref>

Constant rule: if <math>f</math> is constant, then for all Template:Tmath,
<math>f'(x) = 0. </math>
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 Template:Tmath.
Product rule:
<math>(fg)' = f 'g + fg' </math> for all functions <math>f</math> and Template:Tmath. 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 Template:Nowrap.
Chain rule for composite functions: If Template:Tmath, then
<math>f'(x) = h'(g(x)) \cdot g'(x). </math>

Computation exampleEdit

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 Edit

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 Template:Tmath. The derivative of <math> f' </math> is the second derivative, denoted as Template:Tmath, and the derivative of <math> f </math> is the third derivative, denoted as Template:Tmath. By continuing this process, if it exists, the Template:Tmathth derivative is the derivative of the Template:Tmathth derivative or the derivative of order Template:Tmath. As has been discussed above, the generalization of derivative of a function <math> f </math> may be denoted as Template:Tmath.Template:Sfnm A function that has <math> k </math> successive derivatives is called <math> k </math> times differentiable. If the Template:Nowrapth derivative is continuous, then the function is said to be of differentiability class Template:Tmath.Template:Sfn A function that has infinitely many derivatives is called infinitely differentiable or smooth.Template:Sfn 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.Template:Sfn

One 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,Template:Sfn and the third derivative is the jerk.Template:Sfn

In other dimensionsEdit

Template:See also

Vector-valued functionsEdit

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 curves 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, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is,Template:Sfn <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 Template:Tmath, then <math> \mathbf{y}' </math> is another vector-valued function.Template:Sfn

Partial derivativesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Functions can depend upon 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 Template:Block indent among other possibilities.Template:Sfnm It can be thought of as the rate of change of the function in the <math>x</math>-direction.Template:Sfn 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".Template:Sfnm For example, let Template:Tmath, 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:Template:Sfn <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 Template:Tmath, 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.Template:Sfn

Directional derivativesEdit

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If <math> f </math> is a real-valued function on Template:Tmath, 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 Template:Tmath, 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 Template:Tmath. These are measured using directional derivatives. Given a vector Template:Tmath, then the directional derivative of <math> f </math> in the direction of <math> \mathbf{v} </math> at the point <math> \mathbf{x} </math> is:Template:Sfn <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>

If all the partial derivatives of <math> f </math> exist and are continuous at Template:Tmath, then they determine the directional derivative of <math> f </math> in the direction <math> \mathbf{v} </math> by the formula:Template:Sfn <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 matrixEdit

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When <math> f </math> is a function from an open subset of <math> \R^n </math> to Template:Tmath, 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 Template:Tmath, 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 Template:Tmath, the linear approximation formula holds:Template:Sfn <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 thatTemplate:Sfn <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 Template:Tmath, 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 Template:Tmath, and the norm in the numerator is the standard length on <math> \R^m </math>.Template:Sfn If <math> v </math> is a vector starting at Template:Tmath, then <math> f'(\mathbf{a}) \mathbf{v} </math> is called the pushforward of <math> \mathbf{v} </math> by <math> f </math>.Template:Sfn

If the total derivative exists at Template:Tmath, then all the partial derivatives and directional derivatives of <math> f </math> exist at Template:Tmath, and for all Template:Tmath, <math> f'(\mathbf{a})\mathbf{v} </math> is the directional derivative of <math> f </math> in the direction Template:Tmath. If <math> f </math> is written using coordinate functions, so that Template:Tmath, then the total derivative can be expressed using the partial derivatives as a matrix. This matrix is called the Jacobian matrix of <math> f </math> at <math> \mathbf{a} </math>:Template:Sfn <math display="block">f'(\mathbf{a}) = \operatorname{Jac}_{\mathbf{a}} = \left(\frac{\partial f_i}{\partial x_j}\right)_{ij}.</math>

GeneralizationsEdit

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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 functions of complex variables, such as functions from (a domain in) the complex numbers <math>\C</math> to Template:Tmath. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition.Template:Sfn If <math>\C</math> is identified with <math>\R^2</math> by writing a complex number <math>z</math> as Template:Tmath 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 functions.Template:Sfn
Another generalization concerns functions between 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 Template:Tmath. The derivative (or differential) of a (differentiable) map <math>f:M\to N</math> between manifolds, at a point <math>x</math> in Template:Tmath, 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 Template:Tmath. The derivative function becomes a map between the tangent bundles of <math>M</math> and Template:Tmath. This definition is used in differential geometry.Template:Sfn
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>Template:Harvnb. See p. 209 for the Gateaux derivative, and p. 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 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 distributions and only require that a function is differentiable "on average".Template:Sfn
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 rings, ideals, field, and so on.Template:Sfn
The discrete equivalent of differentiation is finite differences. The study of differential calculus is unified with the calculus of finite differences in time scale calculus.Template:Sfn
The arithmetic derivative involves the function that is defined for the integers by the prime factorization. This is an analogy with the product rule.Template:Sfn