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This article is about the term as used in calculus. Derivative_sentence_0

For a less technical overview of the subject, see differential calculus. Derivative_sentence_1

For other uses, see Derivative (disambiguation). Derivative_sentence_2

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivative_sentence_3

Derivatives are a fundamental tool of calculus. Derivative_sentence_4

For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. Derivative_sentence_5

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. Derivative_sentence_6

The tangent line is the best linear approximation of the function near that input value. Derivative_sentence_7

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. Derivative_sentence_8

Derivatives may be generalized to functions of several real variables. Derivative_sentence_9

In this generalization, 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. Derivative_sentence_10

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. Derivative_sentence_11

It can be calculated in terms of the partial derivatives with respect to the independent variables. Derivative_sentence_12

For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. Derivative_sentence_13

The process of finding a derivative is called differentiation. Derivative_sentence_14

The reverse process is called antidifferentiation. Derivative_sentence_15

The fundamental theorem of calculus relates antidifferentiation with integration. Derivative_sentence_16

Differentiation and integration constitute the two fundamental operations in single-variable calculus. Derivative_sentence_17

Differentiation Derivative_section_0

Differentiation is the action of computing a derivative. Derivative_sentence_18

The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. Derivative_sentence_19

It is called the derivative of f with respect to x. Derivative_sentence_20

If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. Derivative_sentence_21

The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y is a line. Derivative_sentence_22

In this case, y = f(x) = mx + b, for real numbers m and b, and the slope m is given by Derivative_sentence_23

The above formula holds because Derivative_sentence_24

Thus Derivative_sentence_25

This gives the value for the slope of a line. Derivative_sentence_26

The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differences Δy / Δx as Δx tends towards 0. Derivative_sentence_27

Notation Derivative_section_1

Main article: Notation for differentiation Derivative_sentence_28

For more detail, see the Notation (details) section. Derivative_sentence_29

Two distinct notations are commonly used for the derivative, one deriving from Gottfried Wilhelm Leibniz and the other from Joseph Louis Lagrange. Derivative_sentence_30

A third notation, first used by Isaac Newton, is sometimes seen in physics. Derivative_sentence_31

In Leibniz's notation, an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written Derivative_sentence_32

suggesting the ratio of two infinitesimal quantities. Derivative_sentence_33

(The above expression is read as "the derivative of y with respect to x", "dy by dx", or "dy over dx". Derivative_sentence_34

The oral form "dy dx" is often used conversationally, although it may lead to confusion.) Derivative_sentence_35

In Lagrange's notation, the derivative with respect to x of a function f(x) is denoted f'(x) (read as "f prime of x") or fx′(x) (read as "f prime x of x"), in case of ambiguity of the variable implied by the differentiation. Derivative_sentence_36

Lagrange's notation is sometimes incorrectly attributed to Newton. Derivative_sentence_37

Newton's notation for differentiation (also called the dot notation for differentiation) places a dot over the dependent variable. Derivative_sentence_38

That is, if y is a function of t, then the derivative of y with respect to t is Derivative_sentence_39

Higher derivatives are represented using multiple dots, as in Derivative_sentence_40

Rigorous definition Derivative_section_2

The most common approach to turn this intuitive idea into a precise definition is to define the derivative as a limit of difference quotients of real numbers. Derivative_sentence_41

This is the approach described below. Derivative_sentence_42

Let f be a real valued function defined in an open neighborhood of a real number a. Derivative_sentence_43

In classical geometry, the tangent line to the graph of the function f at a was the unique line through the point (a, f(a)) that did not meet the graph of f transversally, meaning that the line did not pass straight through the graph. Derivative_sentence_44

The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at (a, f(a)). Derivative_sentence_45

The slope of the tangent line is very close to the slope of the line through (a, f(a)) and a nearby point on the graph, for example (a + h, f(a + h)). Derivative_sentence_46

These lines are called secant lines. Derivative_sentence_47

A value of h close to zero gives a good approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations. Derivative_sentence_48

The slope m of the secant line is the difference between the y values of these points divided by the difference between the x values, that is, Derivative_sentence_49

This expression is Newton's difference quotient. Derivative_sentence_50

Passing from an approximation to an exact answer is done using a limit. Derivative_sentence_51

Geometrically, the limit of the secant lines is the tangent line. Derivative_sentence_52

Therefore, the limit of the difference quotient as h approaches zero, if it exists, should represent the slope of the tangent line to (a, f(a)). Derivative_sentence_53

This limit is defined to be the derivative of the function f at a: Derivative_sentence_54

When the limit exists, f is said to be differentiable at a. Derivative_sentence_55

Here f′(a) is one of several common notations for the derivative (see below). Derivative_sentence_56

From this definition it is obvious that a differentiable function f is increasing if and only if its derivative is positive, and is decreasing iff its derivative is negative. Derivative_sentence_57

This fact is used extensively when analyzing function behavior, e.g. when finding local extrema. Derivative_sentence_58

Equivalently, the derivative satisfies the property that Derivative_sentence_59

which has the intuitive interpretation (see Figure 1) that the tangent line to f at a gives the best linear approximation Derivative_sentence_60

to f near a (i.e., for small h). Derivative_sentence_61

This interpretation is the easiest to generalize to other settings (see below). Derivative_sentence_62

Substituting 0 for h in the difference quotient causes division by zero, so the slope of the tangent line cannot be found directly using this method. Derivative_sentence_63

Instead, define Q(h) to be the difference quotient as a function of h: Derivative_sentence_64

Q(h) is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). Derivative_sentence_65

If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from h = 0. Derivative_sentence_66

If the limit limh→0Q(h) exists, meaning that there is a way of choosing a value for Q(0) that makes Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q(0). Derivative_sentence_67

In practice, the existence of a continuous extension of the difference quotient Q(h) to h = 0 is shown by modifying the numerator to cancel h in the denominator. Derivative_sentence_68

Such manipulations can make the limit value of Q for small h clear even though Q is still not defined at h = 0. Derivative_sentence_69

This process can be long and tedious for complicated functions, and many shortcuts are commonly used to simplify the process. Derivative_sentence_70

Definition over the hyperreals Derivative_section_3

Relative to a hyperreal extension R ⊂ R of the real numbers, the derivative of a real function y = f(x) at a real point x can be defined as the shadow of the quotient ∆y/∆x for infinitesimal ∆x, where ∆y = f(x + ∆x) − f(x). Derivative_sentence_71

Here the natural extension of f to the hyperreals is still denoted f. Here the derivative is said to exist if the shadow is independent of the infinitesimal chosen. Derivative_sentence_72

Example Derivative_section_4

The square function given by f(x) = x is differentiable at x = 3, and its derivative there is 6. Derivative_sentence_73

This result is established by calculating the limit as h approaches zero of the difference quotient of f(3): Derivative_sentence_74

The last expression shows that the difference quotient equals 6 + h when h ≠ 0 and is undefined when h = 0, because of the definition of the difference quotient. Derivative_sentence_75

However, the definition of the limit says the difference quotient does not need to be defined when h = 0. Derivative_sentence_76

The limit is the result of letting h go to zero, meaning it is the value that 6 + h tends to as h becomes very small: Derivative_sentence_77

Hence the slope of the graph of the square function at the point (3, 9) is 6, and so its derivative at x = 3 is f′(3) = 6. Derivative_sentence_78

More generally, a similar computation shows that the derivative of the square function at x = a is f′(a) = 2a: Derivative_sentence_79

Continuity and differentiability Derivative_section_5

If f is differentiable at a, then f must also be continuous at a. Derivative_sentence_80

As an example, choose a point a and let f be the step function that returns the value 1 for all x less than a, and returns a different value 10 for all x greater than or equal to a. f cannot have a derivative at a. Derivative_sentence_81

If h is negative, then a + h is on the low part of the step, so the secant line from a to a + h is very steep, and as h tends to zero the slope tends to infinity. Derivative_sentence_82

If h is positive, then a + h is on the high part of the step, so the secant line from a to a + h has slope zero. Derivative_sentence_83

Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. Derivative_sentence_84

However, even if a function is continuous at a point, it may not be differentiable there. Derivative_sentence_85

For example, the absolute value function given by f(x) = |x| is continuous at x = 0, but it is not differentiable there. Derivative_sentence_86

If h is positive, then the slope of the secant line from 0 to h is one, whereas if h is negative, then the slope of the secant line from 0 to h is negative one. Derivative_sentence_87

This can be seen graphically as a "kink" or a "cusp" in the graph at x = 0. Derivative_sentence_88

Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by f(x) = x is not differentiable at x = 0. Derivative_sentence_89

In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative. Derivative_sentence_90

Most functions that occur in practice have derivatives at all points or at almost every point. Derivative_sentence_91

Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points. Derivative_sentence_92

Under mild conditions, for example if the function is a monotone function or a Lipschitz function, this is true. Derivative_sentence_93

However, in 1872 Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. Derivative_sentence_94

This example is now known as the Weierstrass function. Derivative_sentence_95

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. Derivative_sentence_96

Informally, this means that hardly any random continuous functions have a derivative at even one point. Derivative_sentence_97

The derivative as a function Derivative_section_6

Sometimes f has a derivative at most, but not all, points of its domain. Derivative_sentence_98

The function whose value at a equals f′(a) whenever f′(a) is defined and elsewhere is undefined is also called the derivative of f. It is still a function, but its domain is strictly smaller than the domain of f. Derivative_sentence_99

Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions. Derivative_sentence_100

If we denote this operator by D, then D(f) is the function f′. Derivative_sentence_101

Since D(f) is a function, it can be evaluated at a point a. Derivative_sentence_102

By the definition of the derivative function, D(f)(a) = f′(a). Derivative_sentence_103

For comparison, consider the doubling function given by f(x) = 2x; f is a real-valued function of a real number, meaning that it takes numbers as inputs and has numbers as outputs: Derivative_sentence_104

The operator D, however, is not defined on individual numbers. Derivative_sentence_105

It is only defined on functions: Derivative_sentence_106

Because the output of D is a function, the output of D can be evaluated at a point. Derivative_sentence_107

For instance, when D is applied to the square function, x ↦ x, D outputs the doubling function x ↦ 2x, which we named f(x). Derivative_sentence_108

This output function can then be evaluated to get f(1) = 2, f(2) = 4, and so on. Derivative_sentence_109

Higher derivatives Derivative_section_7

Let f be a differentiable function, and let f ′ be its derivative. Derivative_sentence_110

The derivative of f ′ (if it has one) is written f ′′ and is called the second derivative of f. Similarly, the derivative of the second derivative, if it exists, is written f ′′′ and is called the third derivative of f. Continuing this process, one can define, if it exists, the nth derivative as the derivative of the (n-1)th derivative. Derivative_sentence_111

These repeated derivatives are called higher-order derivatives. Derivative_sentence_112

The nth derivative is also called the derivative of order n. Derivative_sentence_113

If x(t) represents the position of an object at time t, then the higher-order derivatives of x have specific interpretations in physics. Derivative_sentence_114

The first derivative of x is the object's velocity. Derivative_sentence_115

The second derivative of x is the acceleration. Derivative_sentence_116

The third derivative of x is the jerk. Derivative_sentence_117

And finally, the fourth through sixth derivatives of x are snap, crackle, and pop; most applicable to astrophysics. Derivative_sentence_118

A function f need not have a derivative (for example, if it is not continuous). Derivative_sentence_119

Similarly, even if f does have a derivative, it may not have a second derivative. Derivative_sentence_120

For example, let Derivative_sentence_121

On the real line, every polynomial function is infinitely differentiable. Derivative_sentence_122

By standard differentiation rules, if a polynomial of degree n is differentiated n times, then it becomes a constant function. Derivative_sentence_123

All of its subsequent derivatives are identically zero. Derivative_sentence_124

In particular, they exist, so polynomials are smooth functions. Derivative_sentence_125

The derivatives of a function f at a point x provide polynomial approximations to that function near x. Derivative_sentence_126

For example, if f is twice differentiable, then Derivative_sentence_127

in the sense that Derivative_sentence_128

If f is infinitely differentiable, then this is the beginning of the Taylor series for f evaluated at x + h around x. Derivative_sentence_129

Inflection point Derivative_section_8

Main article: Inflection point Derivative_sentence_130

Notation (details) Derivative_section_9

Main article: Notation for differentiation Derivative_sentence_131

Leibniz's notation Derivative_section_10

Main article: Leibniz's notation Derivative_sentence_132

and was once thought of as an infinitesimal quotient. Derivative_sentence_133

Higher derivatives are expressed using the notation Derivative_sentence_134

Leibniz's notation allows one to specify the variable for differentiation (in the denominator), which is relevant in partial differentiation. Derivative_sentence_135

It also can be used to write the chain rule as Derivative_sentence_136

Lagrange's notation Derivative_section_11

To denote the number of derivatives beyond this point, some authors use Roman numerals in superscript, whereas others place the number in parentheses: Derivative_sentence_137

Newton's notation Derivative_section_12

Euler's notation Derivative_section_13

If y = f(x) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x. Euler's notation is then written Derivative_sentence_138

although this subscript is often omitted when the variable x is understood, for instance when this is the only independent variable present in the expression. Derivative_sentence_139

Euler's notation is useful for stating and solving linear differential equations. Derivative_sentence_140

Rules of computation Derivative_section_14

Main article: Differentiation rules Derivative_sentence_141

The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. Derivative_sentence_142

In practice, 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. Derivative_sentence_143

Rules for basic functions Derivative_section_15

Here are the rules for the derivatives of the most common basic functions, where a is a real number. Derivative_sentence_144





Rules for combined functions Derivative_section_16

Here are some of the most basic rules for deducing the derivative of a compound function from derivatives of basic functions. Derivative_sentence_145


  • Constant rule: if f(x) is constant, thenDerivative_item_4_4




Computation example Derivative_section_17

The derivative of the function given by Derivative_sentence_146

is Derivative_sentence_147

Here the second term was computed using the chain rule and third using the product rule. Derivative_sentence_148

The known derivatives of the elementary functions x, x, sin(x), ln(x) and exp(x) = e, as well as the constant 7, were also used. Derivative_sentence_149

In higher dimensions Derivative_section_18

See also: Vector calculus and Multivariable calculus Derivative_sentence_150

Vector-valued functions Derivative_section_19

A vector-valued function y of a real variable sends real numbers to vectors in some vector space R. A vector-valued function can be split up into its coordinate functions y1(t), y2(t), ..., yn(t), meaning that y(t) = (y1(t), ..., yn(t)). Derivative_sentence_151

This includes, for example, parametric curves in R or R. The coordinate functions are real valued functions, so the above definition of derivative applies to them. Derivative_sentence_152

The derivative of y(t) is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. Derivative_sentence_153

That is, Derivative_sentence_154

Equivalently, Derivative_sentence_155

if the limit exists. Derivative_sentence_156

The subtraction in the numerator is the subtraction of vectors, not scalars. Derivative_sentence_157

If the derivative of y exists for every value of t, then y′ is another vector-valued function. Derivative_sentence_158

If e1, ..., en is the standard basis for R, then y(t) can also be written as y1(t)e1 + … + yn(t)en. Derivative_sentence_159

If we assume that the derivative of a vector-valued function retains the linearity property, then the derivative of y(t) must be Derivative_sentence_160

because each of the basis vectors is a constant. Derivative_sentence_161

This generalization is useful, for example, if y(t) is the position vector of a particle at time t; then the derivative y′(t) is the velocity vector of the particle at time t. Derivative_sentence_162

Partial derivatives Derivative_section_20

Main article: Partial derivative Derivative_sentence_163

Suppose that f is a function that depends on more than one variable—for instance, Derivative_sentence_164

f can be reinterpreted as a family of functions of one variable indexed by the other variables: Derivative_sentence_165

In other words, every value of x chooses a function, denoted fx, which is a function of one real number. Derivative_sentence_166

That is, Derivative_sentence_167

Once a value of x is chosen, say a, then f(x, y) determines a function fa that sends y to a + ay + y: Derivative_sentence_168

In this expression, a is a constant, not a variable, so fa is a function of only one real variable. Derivative_sentence_169

Consequently, the definition of the derivative for a function of one variable applies: Derivative_sentence_170

The above procedure can be performed for any choice of a. Derivative_sentence_171

Assembling the derivatives together into a function gives a function that describes the variation of f in the y direction: Derivative_sentence_172

This is the partial derivative of f with respect to y. Derivative_sentence_173

Here is a rounded d called the partial derivative symbol. Derivative_sentence_174

To distinguish it from the letter d, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee". Derivative_sentence_175

In general, the partial derivative of a function f(x1, …, xn) in the direction xi at the point (a1, ..., an) is defined to be: Derivative_sentence_176

In the above difference quotient, all the variables except xi are held fixed. Derivative_sentence_177

That choice of fixed values determines a function of one variable Derivative_sentence_178

and, by definition, Derivative_sentence_179

In other words, the different choices of a index a family of one-variable functions just as in the example above. Derivative_sentence_180

This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives. Derivative_sentence_181

This is fundamental for the study of the functions of several real variables. Derivative_sentence_182

Let f(x1, ..., xn) be such a real-valued function. Derivative_sentence_183

If all partial derivatives ∂f / ∂xj of f are defined at the point a = (a1, ..., an), these partial derivatives define the vector Derivative_sentence_184

which is called the gradient of f at a. Derivative_sentence_185

If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇f that maps the point (a1, ..., an) to the vector ∇f(a1, ..., an). Derivative_sentence_186

Consequently, the gradient determines a vector field. Derivative_sentence_187

Directional derivatives Derivative_section_21

Main article: Directional derivative Derivative_sentence_188

If f is a real-valued function on R, then the partial derivatives of f measure its variation in the direction of the coordinate axes. Derivative_sentence_189

For example, if f is a function of x and y, then its partial derivatives measure the variation in f in the x direction and the y direction. Derivative_sentence_190

They do not, however, directly measure the variation of f in any other direction, such as along the diagonal line y = x. Derivative_sentence_191

These are measured using directional derivatives. Derivative_sentence_192

Choose a vector Derivative_sentence_193

The directional derivative of f in the direction of v at the point x is the limit Derivative_sentence_194

In some cases it may be easier to compute or estimate the directional derivative after changing the length of the vector. Derivative_sentence_195

Often this is done to turn the problem into the computation of a directional derivative in the direction of a unit vector. Derivative_sentence_196

To see how this works, suppose that v = λu. Derivative_sentence_197

Substitute h = k/λ into the difference quotient. Derivative_sentence_198

The difference quotient becomes: Derivative_sentence_199

This is λ times the difference quotient for the directional derivative of f with respect to u. Derivative_sentence_200

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. Derivative_sentence_201

Therefore, Dv(f) = λDu(f). Derivative_sentence_202

Because of this rescaling property, directional derivatives are frequently considered only for unit vectors. Derivative_sentence_203

If all the partial derivatives of f exist and are continuous at x, then they determine the directional derivative of f in the direction v by the formula: Derivative_sentence_204

This is a consequence of the definition of the total derivative. Derivative_sentence_205

It follows that the directional derivative is linear in v, meaning that Dv + w(f) = Dv(f) + Dw(f). Derivative_sentence_206

The same definition also works when f is a function with values in R. The above definition is applied to each component of the vectors. Derivative_sentence_207

In this case, the directional derivative is a vector in R. Derivative_sentence_208

Total derivative, total differential and Jacobian matrix Derivative_section_22

Main article: Total derivative Derivative_sentence_209

When f is a function from an open subset of R to R, then the directional derivative of f in a chosen direction is the best linear approximation to f at that point and in that direction. Derivative_sentence_210

But when n > 1, no single directional derivative can give a complete picture of the behavior of f. The total derivative gives a complete picture by considering all directions at once. Derivative_sentence_211

That is, for any vector v starting at a, the linear approximation formula holds: Derivative_sentence_212

Just like the single-variable derivative, f ′(a) is chosen so that the error in this approximation is as small as possible. Derivative_sentence_213

If n and m are both one, then the derivative f ′(a) is a number and the expression f ′(a)v is the product of two numbers. Derivative_sentence_214

But in higher dimensions, it is impossible for f ′(a) to be a number. Derivative_sentence_215

If it were a number, then f ′(a)v would be a vector in R while the other terms would be vectors in R, and therefore the formula would not make sense. Derivative_sentence_216

For the linear approximation formula to make sense, f ′(a) must be a function that sends vectors in R to vectors in R, and f ′(a)v must denote this function evaluated at v. Derivative_sentence_217

To determine what kind of function it is, notice that the linear approximation formula can be rewritten as Derivative_sentence_218

Notice that if we choose another vector w, then this approximate equation determines another approximate equation by substituting w for v. It determines a third approximate equation by substituting both w for v and a + v for a. Derivative_sentence_219

By subtracting these two new equations, we get Derivative_sentence_220

If we assume that v is small and that the derivative varies continuously in a, then f ′(a + v) is approximately equal to f ′(a), and therefore the right-hand side is approximately zero. Derivative_sentence_221

The left-hand side can be rewritten in a different way using the linear approximation formula with v + w substituted for v. The linear approximation formula implies: Derivative_sentence_222

This suggests that f ′(a) is a linear transformation from the vector space R to the vector space R. In fact, it is possible to make this a precise derivation by measuring the error in the approximations. Derivative_sentence_223

Assume that the error in these linear approximation formula is bounded by a constant times ||v||, where the constant is independent of v but depends continuously on a. Derivative_sentence_224

Then, after adding an appropriate error term, all of the above approximate equalities can be rephrased as inequalities. Derivative_sentence_225

In particular, f ′(a) is a linear transformation up to a small error term. Derivative_sentence_226

In the limit as v and w tend to zero, it must therefore be a linear transformation. Derivative_sentence_227

Since we define the total derivative by taking a limit as v goes to zero, f ′(a) must be a linear transformation. Derivative_sentence_228

In one variable, the fact that the derivative is the best linear approximation is expressed by the fact that it is the limit of difference quotients. Derivative_sentence_229

However, the usual difference quotient does not make sense in higher dimensions because it is not usually possible to divide vectors. Derivative_sentence_230

In particular, the numerator and denominator of the difference quotient are not even in the same vector space: The numerator lies in the codomain R while the denominator lies in the domain R. Furthermore, the derivative is a linear transformation, a different type of object from both the numerator and denominator. Derivative_sentence_231

To make precise the idea that f ′(a) is the best linear approximation, it is necessary to adapt a different formula for the one-variable derivative in which these problems disappear. Derivative_sentence_232

If f : R → R, then the usual definition of the derivative may be manipulated to show that the derivative of f at a is the unique number f ′(a) such that Derivative_sentence_233

This is equivalent to Derivative_sentence_234

because the limit of a function tends to zero if and only if the limit of the absolute value of the function tends to zero. Derivative_sentence_235

This last formula can be adapted to the many-variable situation by replacing the absolute values with norms. Derivative_sentence_236

The definition of the total derivative of f at a, therefore, is that it is the unique linear transformation f ′(a) : R → R such that Derivative_sentence_237

Here h is a vector in R, so the norm in the denominator is the standard length on R. However, f′(a)h is a vector in R, and the norm in the numerator is the standard length on R. If v is a vector starting at a, then f ′(a)v is called the pushforward of v by f and is sometimes written f∗v. Derivative_sentence_238

If the total derivative exists at a, then all the partial derivatives and directional derivatives of f exist at a, and for all v, f ′(a)v is the directional derivative of f in the direction v. If we write f using coordinate functions, so that f = (f1, f2, ..., fm), then the total derivative can be expressed using the partial derivatives as a matrix. Derivative_sentence_239

This matrix is called the Jacobian matrix of f at a: Derivative_sentence_240

The existence of the total derivative f′(a) 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 a. Derivative_sentence_241

The definition of the total derivative subsumes the definition of the derivative in one variable. Derivative_sentence_242

That is, if f is a real-valued function of a real variable, then the total derivative exists if and only if the usual derivative exists. Derivative_sentence_243

The Jacobian matrix reduces to a 1×1 matrix whose only entry is the derivative f′(x). Derivative_sentence_244

This 1×1 matrix satisfies the property that f(a + h) − (f(a) + f ′(a)h) is approximately zero, in other words that Derivative_sentence_245

The total derivative of a function does not give another function in the same way as the one-variable case. Derivative_sentence_246

This is because the total derivative of a multivariable function has to record much more information than the derivative of a single-variable function. Derivative_sentence_247

Instead, the total derivative gives a function from the tangent bundle of the source to the tangent bundle of the target. Derivative_sentence_248

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. Derivative_sentence_249

The analog of a higher-order derivative, called a 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. Derivative_sentence_250

It cannot be a function on the tangent bundle because the tangent bundle only has room for the base space and the directional derivatives. Derivative_sentence_251

Because jets capture higher-order information, they take as arguments additional coordinates representing higher-order changes in direction. Derivative_sentence_252

The space determined by these additional coordinates is called the jet bundle. Derivative_sentence_253

The relation between the total derivative and the partial derivatives of a function is paralleled in the relation between the kth order jet of a function and its partial derivatives of order less than or equal to k. Derivative_sentence_254

By repeatedly taking the total derivative, one obtains higher versions of the Fréchet derivative, specialized to R. The kth order total derivative may be interpreted as a map Derivative_sentence_255

which takes a point x in R and assigns to it an element of the space of k-linear maps from R to R – the "best" (in a certain precise sense) k-linear approximation to f at that point. Derivative_sentence_256

By precomposing it with the diagonal map Δ, x → (x, x), a generalized Taylor series may be begun as Derivative_sentence_257

where f(a) is identified with a constant function, xi − ai are the components of the vector x − a, and (Df)i and (Df)jk are the components of Df and Df as linear transformations. Derivative_sentence_258

Generalizations Derivative_section_23

Main article: Generalizations of the derivative Derivative_sentence_259

The concept of a derivative can be extended to many other settings. Derivative_sentence_260

The common thread is that the derivative of a function at a point serves as a linear approximation of the function at that point. Derivative_sentence_261


  • An important generalization of the derivative concerns complex functions of complex variables, such as functions from (a domain in) the complex numbers C to C. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition. If C is identified with R by writing a complex number z as x + iy, then a differentiable function from C to C is certainly differentiable as a function from R to R (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.Derivative_item_8_8
  • Another generalization concerns functions between differentiable or smooth manifolds. Intuitively speaking such a manifold M is a space that can be approximated near each point x by a vector space called its tangent space: the prototypical example is a smooth surface in R. The derivative (or differential) of a (differentiable) map f: M → N between manifolds, at a point x in M, is then a linear map from the tangent space of M at x to the tangent space of N at f(x). The derivative function becomes a map between the tangent bundles of M and N. This definition is fundamental in differential geometry and has many uses – see pushforward (differential) and pullback (differential geometry).Derivative_item_8_9
  • Differentiation can also be defined for maps between infinite dimensional vector spaces such as Banach spaces and Fréchet spaces. There is a generalization both of the directional derivative, called the Gateaux derivative, and of the differential, called the Fréchet derivative.Derivative_item_8_10
  • 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".Derivative_item_8_11
  • The properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology — see, for example, differential algebra.Derivative_item_8_12
  • 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.Derivative_item_8_13
  • Also see arithmetic derivative.Derivative_item_8_14

History Derivative_section_24

Main article: History of calculus Derivative_sentence_262

Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Derivative_sentence_263

Isaac Newton and Gottfried Leibniz independently discovered calculus in the mid-17th century. Derivative_sentence_264

However, each inventor claimed the other stole his work in a bitter dispute that continued until the end of their lives. Derivative_sentence_265

See also Derivative_section_25

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