Derivative

For other uses, see Derivative (disambiguation).

For a non-technical overview of the subject, see Differential calculus. Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change

The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.

In calculus, a branch of mathematics, the derivative is a measurement of how a function changes when the values of its inputs change. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics, matrix calculus is a specialized notation for doing Multivariable calculus, especially over spaces of matrices, where it defines the In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable In Calculus, the quotient rule is a method of finding the Derivative of a function that is the Quotient of two other functions for which In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. In Mathematics, an implicit function is a generalization for the concept of a function in which the Dependent variable has not been given "explicitly" In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor In Differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change The primary operation in Differential calculus is finding a Derivative. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space See the following pages for lists of Integrals: List of integrals of rational functions List of integrals of irrational functions In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other Disk integration is a means of calculating the Volume of a Solid of revolution, when integrating along the axis of revolution Shell integration (the shell method in Integral calculus) is a means of calculating the Volume of a Solid of revolution, when integrating In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires In Mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions In Integral calculus, the use of Partial fractions is required to integrate the general Rational function. In Calculus, interchange of the order of integration is a methodology that transforms multiple integrations of functions into other hopefully simpler integrals by Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point. For example, the derivative of the position or distance of a car at some point in time is the instantaneous velocity, or instantaneous speed (respectively), at which that car is traveling (conversely the integral of the velocity is the car's position). The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

A closely related notion is the differential of a function. In mathematics and more specifically in Differential calculus, the term differential has several interrelated meanings

The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. In Mathematics, a linear approximation is an approximation of a general function using a Linear function (more precisely an Affine function For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In Mathematics, a function of a real variable is a Mathematical function whose domain is the Real line. Slope is used to describe the steepness incline gradient or grade of a straight line. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics and its applications linearization refers to finding the Linear approximation to a function at a given point ^[1]

The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

Differentiation and the derivative

At each point, the derivative is the slope of a line that is tangent to the curve. The red line is always tangent to the blue curve; its slope is the derivative.

At each point, the derivative is the slope of a line that is tangent to the curve. Slope is used to describe the steepness incline gradient or grade of a straight line. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object The red line is always tangent to the blue curve; its slope is the derivative.

Differentiation is a method to compute the rate at which a quantity, y, changes with respect to the change in another quantity, x, upon which it is dependent. Dependent variables and independent variables refer to values that change in relationship to each other This rate of change is called the derivative of y with respect to x. In more precise language, the dependency of y on x means that y is a function of x. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point. In Mathematics, the real numbers may be described informally in several different ways In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) Slope is used to describe the steepness incline gradient or grade of a straight line. This functional relationship is often denoted y = f(x), where f denotes the function.

The simplest case is when y is a linear function of x, meaning that the graph of y against x is a straight line. In Mathematics, the term linear function can refer to either of two different but related concepts In this case, y = f(x) = m x + c, for real numbers m and c, and the slope m is given by

$m={\mbox{change in } y \over \mbox{change in } x} = {\Delta y \over{\Delta x}}$

where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in. Delta (uppercase Δ, lowercase δ; Δέλτα Thelta is the fourth letter of the Greek alphabet. " This formula is true because

y + Δy = f(x+ Δx) = m (x + Δx) + c = m x + c + m Δx = y + mΔx.

It follows that Δy = m Δx.

This gives an exact value for the slope of a straight line. If the function f is not linear (i. e. its graph is not a straight line), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

Figure 1. The tangent line at (x, f(x))

Figure 2. The secant to curve y= f(x) determined by points (x, f(x)) and (x+h, f(x+h)).

Figure 2. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. The secant to curve y= f(x) determined by points (x, f(x)) and (x+h, f(x+h)). A secant line of a Curve is a line that (locally intersects two points on the curve

Figure 3. The tangent line as limit of secants.

The idea, illustrated by Figures 1-3, is to compute the rate of change as the limiting value of the ratio of the differences Δy / Δx as Δx becomes infinitely small. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular The primary vehicle of Calculus and other higher mathematics is the function.

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

$\frac{dy}{dx} \,\!$

suggesting the ratio of two infinitesimal quantities. In Calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have (The above expression is pronounced in various ways such as "d y by d x" or "d y over d x". The oral form "d y d x" is often used conversationally, although it may lead to confusion. )

The most common approach^[2] to turn this intuitive idea into a precise definition uses limits, but there are other methods, such as non-standard analysis. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" Non-standard analysis is a branch of Mathematics that formulates analysis using a rigorous notion of an Infinitesimal number ^[3]

Definition via difference quotients

Let y=f(x) be a function of x. In classical geometry, the tangent line at a real number a was the unique line through the point (a, f(a)) which did not meet the graph of f transversally, meaning that the line did not pass straight through the graph. Transversality in Mathematics is a notion that describes how spaces can intersect transversality can be seen as the "opposite" of tangency, and plays a 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. 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)). These lines are called secant lines. A secant line of a Curve is a line that (locally intersects two points on the curve A value of h close to zero will give a good approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. An approximation (represented by the symbol ≈ is an inexact representation of something that is still close enough to be useful The slope of the secant line is the difference between the y values of these points divided by the difference between the x values, that is,

$\frac{f(a+h)-f(a)}{h}.$

This expression is Newton's difference quotient. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements The primary vehicle of Calculus and other higher mathematics is the function. The derivative is the value of the difference quotient as the secant lines get closer and closer to the tangent line. Formally, the derivative of the function f at a is the limit

$f'(a)=\lim_{h\to 0}{f(a+h)-f(a)\over h}$

of the difference quotient as h approaches zero, if this limit exists. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" If the limit exists, then f is differentiable at a. Here f′ (a) is one of several common notations for the derivative (see below). In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

Equivalently, the derivative satisfies the property that

$\lim_{h\to 0}{f(a+h)-f(a) - f'(a)\cdot h\over h} = 0,$

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

$f(a+h) \approx f(a) + f'(a)h$

to f near a (i. The word linear comes from the Latin word linearis, which means created by lines. e. , for small h). This interpretation is the easiest to generalize to other settings (see below). In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

Substituting 0 for h in the difference quotient causes division by zero, so the slope of the tangent line cannot be found directly. Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric In Instead, define Q(h) to be the difference quotient as a function of h:

$Q(h) = \frac{f(a + h) - f(a)}{h}$ .

Q(h) is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). 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 the point h = 0. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output If the limit $\textstyle\lim_{h\to 0} Q(h)$ exists, meaning that there is a way of choosing a value for Q(0) which makes the graph of Q a continuous function, then the function f is differentiable at the point a, and its derivative at a equals Q(0).

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

Example

The squaring function f(x) = x² is differentiable at x = 3, and its derivative there is 6. This is proven by writing the difference quotient as follows:

${f(3+h)-f(3)\over h} = {(3+h)^2 - 9\over{h}} = {9 + 6h + h^2 - 9\over{h}} = {6h + h^2\over{h}} = 6 + h.$

Then we get the simplified function in the limit:

$\lim_{h\to 0} 6 + h = 6 + 0 = 6.$

The last expression shows that the difference quotient equals 6 + h when h is not zero and is undefined when h is zero. (Remember that because of the definition of the difference quotient, the difference quotient is always undefined when h is zero. ) However, there is a natural way of filling in a value for the difference quotient at zero, namely 6. Hence the slope of the graph of the squaring function at the point (3, 9) is 6, and so its derivative at x = 3 is f '(3) = 6.

More generally, a similar computation shows that the derivative of the squaring function at x = a is f '(a) = 2a.

Continuity and differentiability

This function does not have a derivative at the marked point, as the function is not continuous there.

If y = f(x) is differentiable at a, then f must also be continuous at a. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output As an example, choose a point a and let f be the step function which returns a value, say 1, for all x less than a, and returns a different value, say 10, for all x greater than or equal to a. In Mathematics, a function on the Real numbers is called a step function (or staircase function) if it can be written as a finite f cannot have a derivative at a. If h is negative, then a + h is on the low part of the step, so the secant line from a to a + h will be very steep, and as h tends to zero the slope tends to infinity. If h is positive, then a + h is on the high part of the step, so the secant line from a to a + h will have slope zero. Consequently the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. ^[4]

The absolute value function is continuous, but fails to be differentiable at x = 0 since it has a sharp corner.

However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolute value function y = |x| is continuous at x = 0, but it is not differentiable there. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. 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. This can be seen graphically as a "kink" in the graph at x = 0. Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function y = ³√x is not differentiable at x = 0. In Mathematics, a vertical tangent is Tangent line with Infinite Slope, thus being Vertical.

Most functions which occur in practice have derivatives at all points or at almost every point. In Measure theory (a branch of Mathematical analysis) one says that a property holds almost everywhere if the set of elements for which the property does However, a result of Stefan Banach states that the set of functions which have a derivative at some point is a meager set in the space of all continuous functions. Stefan Banach ( Ukrainian: Степан Степанович Банах 1892–1945 was a Polish Mathematician who worked in interwar Poland and in In the mathematical fields of General topology and Descriptive set theory, a meagre set (also called a meager set or a set of first category) ^[5] Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function. In Mathematics, the Weierstrass function is a pathological example of a real -valued function on the Real line.

The derivative as a function

Let f be a function that has a derivative at every point a in the domain of f. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined Because every point a has a derivative, there is a function which sends the point a to the derivative of f at a. This function is written f′(x) and is called the derivative function or the derivative of f. The derivative of f collects all the derivatives of f at all the points in the domain of f.

Sometimes f has a derivative at most, but not all, points of its domain. The function whose value at a equals f′(a) whenever f′(a) is defined and is undefined elsewhere is also called the derivative of f. It is still a function, but its domain is strictly smaller than the domain of f.

Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions which have derivatives at every point of their domain and whose range is a set of functions. In Mathematics, an operator is a function which operates on (or modifies another function If we denote this operator by D, then D(f) is the function f′(x). Since D(f) is a function, it can be evaluated at a point a. By the definition of the derivative function, D(f)(a) = f′(a).

For comparison, consider the doubling function 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:

$\begin{align} 1 &{}\mapsto 2,\\ 2 &{}\mapsto 4,\\ 3 &{}\mapsto 6. \end{align}$

The operator D, however, is not defined on individual numbers. It is only defined on functions:

$\begin{align} D(x \mapsto 1) &= (x \mapsto 0),\\ D(x \mapsto x) &= (x \mapsto 1),\\ D(x \mapsto x^2) &= (x \mapsto 2\cdot x). \end{align}$

Because the output of D is a function, the output of D can be evaluated at a point. For instance, when D is applied to the squaring function,

$x \mapsto x^2,$

D outputs the doubling function,

$x \mapsto 2x ,$

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

Higher derivatives

Let f be a differentiable function, and let f′(x) be its derivative. The derivative of f′(x) (if it has one) is written f′′(x) and is called the second derivative of f. Similarly, the derivative of a second derivative, if it exists, is written f′′′(x) and is called the third derivative of f. These repeated derivatives are called higher-order derivatives.

A function f need not have a derivative, for example, if it is not continuous. Similarly, even if f does have a derivative, it may not have a second derivative. For example, let

$f(x) = \begin{cases} x^2, & \mbox{if }x\ge 0 \\ -x^2, & \mbox{if }x \le 0\end{cases}$ .

An elementary calculation shows that f is a differentiable function whose derivative is

$f'(x) = \begin{cases} 2x, & \mbox{if }x\ge 0 \\ -2x, & \mbox{if }x \le 0\end{cases}$ .

f′(x) is twice the absolute value function, and it does not have a derivative at zero. Similar examples show that a function can have k derivatives for any non-negative integer k but no (k + 1)-order derivative. A function that has k successive derivatives is called k times differentiable. If in addition the kth derivative is continuous, then the function is said to be of differentiability class C^k. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability (This is a stronger condition than having k derivatives. For an example, see differentiability class. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability ) A function that has infinitely many derivatives is called infinitely differentiable or smooth. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability

On the real line, every polynomial function is infinitely differentiable. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations By standard differentiation rules, if a polynomial of degree n is differentiated n times, then it becomes a constant function. This is a summary of differentiation rules, that is rules for computing the Derivative of a function in Calculus. In Mathematics, a constant function is a function whose values do not vary and thus are Constant. All of its subsequent derivatives are identically zero. In particular, they exist, so polynomials are smooth functions.

The derivatives of a function f at a point x provide polynomial approximations to that function near x. For example, if f is twice differentiable, then

$f(x+h) \approx f(x) + f'(x)h + \tfrac12 f''(x) h^2$

in the sense that

$\lim_{h\to 0}\frac{f(x+h) - f(x) - f'(x)h - \frac12 f''(x) h^2}{h^2}=0.$

If f is infinitely differentiable, then this is the beginning of the Taylor series for f. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

Notations for differentiation

Main article: Notation for differentiation

Leibniz's notation

Main article: Leibniz's notation

The notation for derivatives introduced by Gottfried Leibniz is one of the earliest. There is no single uniform notation for differentiation. Instead several different notations for the Derivative of a function or variable have been proposed In Calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz It is still commonly used when the equation y=f(x) is viewed as a functional relationship between dependent and independent variables. Dependent variables and independent variables refer to values that change in relationship to each other Then the first derivative is denoted by

$\frac{dy}{dx},\quad\frac{d f}{dx}(x),\;\;\mathrm{or}\;\; \frac{d}{dx}f(x).$

Higher derivatives are expressed using the notation

$\frac{d^ny}{dx^n}, \quad\frac{d^nf}{dx^n}(x), \;\;\mathrm{or}\;\; \frac{d^n}{dx^n}f(x)$

for the nth derivative of y = f(x) (with respect to x).

With Leibniz's notation, we can write the derivative of y at the point x = a in two different ways:

$\frac{dy}{dx}\left.{\!\!\frac{}{}}\right|_{x=a} = \frac{dy}{dx}(a).$

Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant It also makes the chain rule easy to remember:^[6]

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.$

Lagrange's notation

One of the most common modern notations for differentiation is due to Joseph Louis Lagrange and uses the prime mark, so that the derivative of a function f(x) is denoted f′(x) or simply f′. In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. The prime symbol ( ′ double prime symbol ( &Prime triple prime symbol ( ‴ etc Similarly, the second and third derivatives are denoted

$(f')'=f''\,$ and $(f'')'=f'''\,.$

Beyond this point, some authors use Roman numerals such as

$f^{\mathrm{iv}}\,$

for the fourth derivative, whereas other authors place the number of derivatives in parentheses:

$f^{(4)}\,$

The latter notation generalizes to yield the notation f ⁽ⁿ⁾ for the nth derivative of f — this notation is most useful when we wish to talk about the derivative as being a function itself, as in this case the Leibniz notation can become cumbersome.

Newton's notation

Main article: Newton's notation

Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a derivative. Newton's notation for differentiation involved placing a dash/dot over the function name which he termed the fluxion. Newton's notation for differentiation involved placing a dash/dot over the function name which he termed the fluxion. If y = f(t), then

$\dot{y}$ and $\ddot{y}$

denote, respectively, the first and second derivatives of y with respect to t. This notation is used almost exclusively for time derivatives, meaning that the independent variable of the function represents time. A time derivative is a Derivative of a function with respect to Time, usually interpreted as the Rate of change of the value of the function For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of It is very common in physics and in mathematical disciplines connected with physics such as differential equations. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the While the notation becomes unmanageable for high-order derivatives, in practice only very few derivatives are needed.

Euler's notation

Euler's notation uses a differential operator D, which is applied to a function f to give the first derivative Df. In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator The second derivative is denoted D²f, and the nth derivative is denoted Dⁿf.

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

$D_x y\,$ or $D_x f(x)\,$ ,

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

Euler's notation is useful for stating and solving linear differential equations. In Mathematics, a linear differential equation is a Differential equation of the form Ly = f \ where the Differential

Computing the derivative

The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. For some examples, see Derivative (examples). Example 1 Consider f ( x) = 5 f'(x=\lim_{h\rightarrow 0} \frac{f(x+h-f(x}{h} = \lim_{h\rightarrow 0} \frac{f(x+h-5}{h} = \lim_{h\rightarrow 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.

Derivatives of elementary functions

Main article: Table of derivatives

In addition, the derivatives of some common functions are useful to know. The primary operation in Differential calculus is finding a Derivative.

Derivatives of powers: if

$f(x) = x^r\,$ ,

where r is any real number, then

$f'(x) = rx^{r-1}\,$ ,

wherever this function is defined. In Mathematics, Polynomials are perhaps the simplest functions with which to do Calculus. In Mathematics, the real numbers may be described informally in several different ways For example, if r = 1/2, then

$f'(x) = \frac{1}{2}x^{-\tfrac12}\,$ .

and the function is defined only for non-negative x. When r = 0, this rule recovers the constant rule.

Exponential and logarithm functions:

$\frac{d}{dx}e^x = e^x$

$\frac{d}{dx}a^x = \ln(a)a^x$

$\frac{d}{dx}\ln(x) = 1/x,\qquad x > 0$

$\frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}$

Trigonometric functions:

$\frac{d}{dx}\sin(x) = \cos(x).$

$\frac{d}{dx}\cos(x)= -\sin(x).$

$\frac{d}{dx}\tan(x)= \sec^2(x).$

Inverse trigonometric functions:

$\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}}.$

$\frac{d}{dx}\arccos(x)= -\frac{1}{\sqrt{1-x^2}}.$

$\frac{d}{dx}\arctan(x)= \frac{1}{{1+x^2}}.$

Rules for finding the derivative

Main article: Differentiation rules

In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided using differentiation rules. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce This is a summary of differentiation rules, that is rules for computing the Derivative of a function in Calculus. Some of the most basic rules are the following.

Constant rule: if f(x) is constant, then

$f' = 0 \,$

Sum rule:

$(af + bg)' = af' + bg' \,$ for all functions f and g and all real numbers a and b. In Mathematics, the linearity of differentiation is a most fundamental property of the Derivative, in Differential calculus.

Product rule:

$(fg)' = f 'g + fg' \,$ for all functions f and g. In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable

Quotient rule:

$\left(\frac{f}{g} \right)' = \frac{f'g - fg'}{g^2}$ for all functions f and g where g ≠ 0. In Calculus, the quotient rule is a method of finding the Derivative of a function that is the Quotient of two other functions for which

Chain rule: If $f (x) = h (g (x))$ , then

$f'(x) = h'(g(x)) \cdot g'(x) \,$ . In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions.

Example computation

The derivative of

$f(x) = x^4 + \sin (x^2) - \ln(x) e^x + 7\,$

$\begin{align} f'(x) &= 4 x^{(4-1)}+ \frac{d\left(x^2\right)}{dx}\cos (x^2) - \frac{d\left(\ln {x}\right)}{dx} e^x - \ln{x} \frac{d\left(e^x\right)}{dx} + 0 \\ &= 4x^3 + 2x\cos (x^2) - \frac{1}{x} e^x - \ln(x) e^x. \end{align}$

Here the second term was computed using the chain rule and third using the product rule: the known derivatives of the elementary functions x², x⁴, sin(x), ln(x) and exp(x) = e^x were also used.

Derivatives in higher dimensions

See also: vector calculus and multivariable calculus

Derivatives of vector valued functions

A vector-valued function y(t) of a real variable is a function which sends real numbers to vectors in some vector space Rⁿ. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner Multivariable calculus is the extension of Calculus in one Variable to calculus in several variables the functions which are differentiated and integrated involve A vector-valued function is a mathematical function that maps Real numbers onto vectors Vector-valued functions can be defined as \mathbf{r}(t=f(t\mathbf+g(t\mathbf In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added A vector-valued function can be split up into its coordinate functions y₁(t), y₂(t), …, y_n(t), meaning that y(t) = (y₁(t), . . . , y_n(t)). This includes, for example, parametric curves in R² or R³. This article only considers curves in Euclidean space Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian The coordinate functions are real valued functions, so the above definition of derivative applies to them. The derivative of y(t) is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is,

$\mathbf{y}'(t) = (y'_1(t), \ldots, y'_n(t)).$

Equivalently,

$\mathbf{y}'(t)=\lim_{h\to 0}\frac{\mathbf{y}(t+h) - \mathbf{y}(t)}{h},$

if the limit exists. The subtraction in the numerator is subtraction of vectors, not scalars. If the derivative of y exists for every value of t, then y′ is another vector valued function.

If e₁, …, e_n is the standard basis for Rⁿ, then y(t) can also be written as y₁(t)e₁ + … + y_n(t)e_n. If we assume that the derivative of a vector-valued function retains the linearity property, then the derivative of y(t) must be

$y'_1(t)\mathbf{e}_1 + \cdots + y'_n(t)\mathbf{e}_n$

because each of the basis vectors is a constant. In Mathematics, the linearity of differentiation is a most fundamental property of the Derivative, in Differential calculus.

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. In Physics, velocity is defined as the rate of change of Position.

Partial derivatives

Main article: Partial derivative

Suppose that f is a function that depends on more than one variable. In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant For instance,

$f(x,y) = x^2 + xy + y^2.\,$

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

$f(x,y) = f_x(y) = x^2 + xy + y^2.\,$

In other words, every value of x chooses a function, denoted f_x, which is a function of one real number. ^[7] That is,

$x \mapsto f_x,\,$

$f_x(y) = x^2 + xy + y^2.\,$

Once a value of x is chosen, say a, then f(x,y) determines a function f_a which sends y to a² + ay + y²:

$f_a(y) = a^2 + ay + y^2.\,$

In this expression, a is a constant, not a variable, so f_a is a function of only one real variable. Consequently the definition of the derivative for a function of one variable applies:

$f_a'(y) = a + 2y.\,$

The above procedure can be performed for any choice of a. Assembling the derivatives together into a function gives a function which describes the variation of f in the y direction:

$\frac{\part f}{\part y}(x,y) = x + 2y.$

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

In general, the partial derivative of a function f(x₁, …, x_n) in the direction x_i at the point (a₁ …, a_n) is defined to be:

$\frac{\part f}{\part x_i}(a_1,\ldots,a_n) = \lim_{h \to 0}\frac{f(a_1,\ldots,a_i+h,\ldots,a_n) - f(a_1,\ldots,a_n)}{h}.$

In the above difference quotient, all the variables except x_i are held fixed. That choice of fixed values determines a function of one variable

$f_{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n}(x_i) = f(a_1,\ldots,a_{i-1},x_i,a_{i+1},\ldots,a_n)$

and, by definition,

$\frac{df_{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n}}{dx_i}(a_1,\ldots,a_n) = \frac{\part f}{\part x_i}(a_1,\ldots,a_n).$

In other words, the different choices of a index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.

An important example of a function of several variables is the case of a scalar-valued function f(x₁,. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point . . x_n) on a domain in Euclidean space Rⁿ (e. g. , on R² or R³). In this case f has a partial derivative ∂f/∂x_j with respect to each variable x_j. At the point a, these partial derivatives define the vector

$\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \ldots, \frac{\partial f}{\partial x_n}(a)\right).$

This vector is called the gradient of f at a. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇f which takes the point a to the vector ∇f(a). Consequently the gradient determines a vector field. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space.

Directional derivatives

Main article: Directional derivative

If f is a real-valued function on Rⁿ, then the partial derivatives of f measure its variation in the direction of the coordinate axes. In Mathematics, the directional derivative of a multivariate Differentiable function along a given vector V at a given point P intuitively represents the 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. They do not, however, directly measure the variation of f in any other direction, such as along the diagonal line y = x. These are measured using directional derivatives. Choose a vector

$\mathbf{v} = (v_1,\ldots,v_n).$

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

$D_{\mathbf{v}}{f}(\boldsymbol{x}) = \lim_{h \rightarrow 0}{\frac{f(\boldsymbol{x} + h\mathbf{v}) - f(\boldsymbol{x})}{h}}.$

Let λ be a scalar. The substitution of h/λ for h changes the λv direction's difference quotient into λ times the v direction's difference quotient. Consequently, the directional derivative in the λv direction is λ times the directional derivative in the v direction. Because of this, directional derivatives are often considered only for unit vectors v.

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:

$D_{\mathbf{v}}{f}(\boldsymbol{x}) = \sum_{j=1}^n v_j \frac{\partial f}{\partial x_j}.$

This is a consequence of the definition of the total derivative. In the mathematical field of Differential calculus, the term total derivative has a number of closely related meanings It follows that the directional derivative is linear in v. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

The same definition also works when f is a function with values in R^m. We just use the above definition in each component of the vectors. In this case, the directional derivative is a vector in R^m.

The total derivative, the total differential and the Jacobian

Main article: Total derivative

Let f be a function from a domain in R to R. In the mathematical field of Differential calculus, the term total derivative has a number of closely related meanings The derivative of f at a point a in its domain is the best linear approximation to f at that point. As above, this is a number. Geometrically, if v is a unit vector starting at a, then f′ (a) , the best linear approximation to f at a, should be the length of the vector found by moving v to the target space using f. (This vector is called the pushforward of v by f and is usually written $f * v$ . Suppose that &phi: M → N is a smooth map between smooth manifolds then the differential of &phi at a point x is in some ) In other words, if v is measured in terms of distances on the target, then, because v can only be measured through f, v no longer appears to be a unit vector because f does not preserve unit vectors. Instead v appears to have length f′ (a). If m is greater than one, then by writing f using coordinate functions, the length of v in each of the coordinate directions can be measured separately.

Suppose now that f is a function from a domain in Rⁿ to R^m and that a is a point in the domain of f. The derivative of f at a should still be the best linear approximation to f at a. In other words, if v is a vector on Rⁿ, then f′ (a) should be the linear transformation that best approximates f. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that The linear transformation should contain all the information about how f transforms vectors at a to vectors at f(a), and in symbols, this means it should be the linear transformation f′ (a) such that

$\lim_{||\mathbf{h}||\rightarrow 0} \frac{||f(\mathbf{a}+\mathbf{h}) - f(\mathbf{a}) - f'(\mathbf{a})\mathbf{h}||}{||\mathbf{h}||} = 0.$

Here h is a vector in Rⁿ, so the norm in the denominator is the standard length on Rⁿ. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length However, f′ (a)h is a vector in R^m, and the norm in the numerator is the standard length on R^m. The linear transformation f′ (a), if it exists, is called the total derivative of f at a or the (total) differential of f at a. In mathematics and more specifically in Differential calculus, the term differential has several interrelated meanings

If the total derivative exists at a, then all the partial derivatives of f exist at a. If we write f using coordinate functions, so that f = (f₁, f₂, . . . , f_m), then the total derivative can be expressed as a matrix called the Jacobian matrix of f at a:

$f'(\mathbf{a}) = \text{Jac}_{\mathbf{a}} = \left(\frac{\partial f_i}{\partial x_j}\right)_{ij}.$

The existence of the Jacobian is strictly stronger than existence of all the partial derivatives, but if the partial derivatives exist and satisfy mild smoothness conditions, then the total derivative exists and is given by the Jacobian. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant.

The definition of the total derivative subsumes the definition of the derivative in one variable. In this case, the total derivative exists if and only if the usual derivative exists. The Jacobian matrix reduces to a 1×1 matrix whose only entry is the derivative f′ (x). This 1×1 matrix satisfies the property that f(a + h) − f(a) − f′(a)h is approximately zero, in other words that

$f(a+h) \approx f(a) + f'(a)h.$

Up to changing variables, this is the statement that the function $x \mapsto f(a) + f'(a)(x-a)$ is the best linear approximation to f at a.

The total derivative of a function does not give another function in the same way that 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. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the

Generalizations

Main article: Derivative (generalizations)

The concept of a derivative can be extended to many other settings. Derivative is a fundamental construction of Differential calculus and admits many possible generalizations within the fields of Mathematical analysis, Combinatorics The common thread is that the derivative of a function at a point serves as a linear approximation of the function at that point. In Mathematics, a linear approximation is an approximation of a general function using a Linear function (more precisely an Affine function

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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition. However, this innocent definition hides some very deep properties. If C is identified with R² by writing a complex number z as x + i y, then a differentiable function from C to C is certainly differentiable as a function from R² to R² (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. In Mathematics, the Cauchy-Riemann differential equations in Complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane

Another generalization concerns functions between differentiable or smooth manifolds. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. Intuitively speaking such a manifold M is a space which can be approximated near each point x by a vector space called its tangent space: the prototypical example is a smooth surface in R³. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Mathematics, the differential geometry of surfaces deals with smooth Surfaces with various additional structures most often a Riemannian metric 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). In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that The derivative function becomes a map between the tangent bundles of M and N. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the This definition is fundamental in differential geometry and has many uses — see pushforward (differential) and pullback (differential geometry). Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Suppose that &phi: M → N is a smooth map between smooth manifolds then the differential of &phi at a point x is in some Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from

Differentiation can also be defined for maps between infinite dimensional vector spaces such as Banach spaces and Fréchet spaces. In Mathematics, the dimension of a Vector space V is the cardinality (i In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis This article deals with Fréchet spaces in functional analysis There is a generalization both of the directional derivative, called the Gâteaux derivative, and of the differential, called the Fréchet derivative. In Mathematics, the Gâteaux differential is a generalisation of the concept of Directional derivative in Differential calculus. In Mathematics, the Fréchet derivative is a Derivative defined on Banach spaces Named after Maurice Fréchet, it is commonly used to formalize

One deficiency of the classical derivative is that not very many functions are 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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, a weak derivative is a generalization of the concept of the Derivative of a function ( strong derivative) for functions not assumed 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". In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions

The properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology — see, for example, differential algebra. In Mathematics, differential rings differential fields and differential algebras are rings, fields and algebras equipped with a derivation,

Also see arithmetic derivative. In Number theory, the arithmetic derivative, or number derivative, is a function defined for Integers based on Prime factorization, by analogy

Notes

^ Differential calculus, as discussed in this article, is a very well-established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Apostol 1967, Apostol 1969, and Spivak 1994.
^ Spivak 1994, chapter 10.
^ See Differential (infinitesimal) for an overview. In differential calculus, a differential is traditionally an Infinitesimally small change in a Variable. Further approaches include the Radon-Nikodym theorem, and the universal derivation (see Kähler differential). In Mathematics, the Radon–Nikodym theorem is a result in Functional analysis that states that given a measurable space ( X,&Sigma if a In Mathematics, Kähler differentials provide a generalization of Differential forms to arbitrary Commutative rings (or schemes.
^ Despite this, it is still possible to take the derivative in the sense of distributions. In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions The result is nine times the Dirac measure centered at a. In Mathematics, a Dirac measure is a measure &delta x on a set X (with any ''&sigma''-algebra of Subsets
^ Banach, S. (1931). "Uber die Baire'sche Kategorie gewisser Funktionenmengen". Studia. Math. (3): pp. 174–179. . Cited by Hewitt, E and Stromberg, K (1963). Real and abstract analysis. Springer-Verlag, Theorem 17. 8.
^ In the formulation of calculus in terms of limits, the du symbol has been assigned various meanings by various authors. Some authors do not assign a meaning to du by itself, but only as part of the symbol du/dx. Others define "dx" as an independent variable, and define du by du = dx•f′ (x). In non-standard analysis du is defined as an infinitesimal. Non-standard analysis is a branch of Mathematics that formulates analysis using a rigorous notion of an Infinitesimal number It is also interpreted as the exterior derivative du of a function u. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms See differential (infinitesimal) for further information. In differential calculus, a differential is traditionally an Infinitesimally small change in a Variable.
^ This can also be expressed as the adjointness between the product space and function space constructions. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y.

References

Print

Anton, Howard; Bivens, Irl & Davis, Stephen (February 2, 2005), Calculus: Early Transcendentals Single and Multivariable (8th ed. ), New York: Wiley, ISBN 978-0471472445
Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, vol. 1 (2nd ed. ), Wiley, ISBN 978-0471000051
Apostol, Tom M. (June 1969), Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications, vol. 1 (2nd ed. ), Wiley, ISBN 978-0471000075
Eves, Howard (January 2, 1990), An Introduction to the History of Mathematics (6th ed. ), Brooks Cole, ISBN 978-0030295584
Larson, Ron; Hostetler, Robert P. & Edwards, Bruce H. (February 28, 2006), Calculus: Early Transcendental Functions (4th ed. ), Houghton Mifflin Company, ISBN 978-0618606245
Spivak, Michael (September 1994), Calculus (3rd ed. Michael David Spivak (*1940 in Queens, New York) is a Mathematician specializing in Differential geometry, an expositor of Mathematics ), Publish or Perish, ISBN 978-0914098898
Stewart, James (December 24, 2002), Calculus (5th ed. ), Brooks Cole, ISBN 978-0534393397
Thompson, Silvanus P. (September 8, 1998), Calculus Made Easy (Revised, Updated, Expanded ed. ), New York: St. Martin's Press, ISBN 978-0312185480

Online books

Crowell, Benjamin (2003), Calculus, <http://www.lightandmatter.com/calc/>
Garrett, Paul (2004), Notes on First-Year Calculus, <http://www.math.umn.edu/~garrett/calculus/>
Hussain, Faraz (2006), Understanding Calculus, <http://www.understandingcalculus.com/>
Keisler, H. Jerome (2000), Elementary Calculus: An Approach Using Infinitesimals, <http://www.math.wisc.edu/~keisler/calc.html>
Mauch, Sean (2004), Unabridged Version of Sean's Applied Math Book, <http://www.its.caltech.edu/~sean/book/unabridged.html>
Sloughter, Dan (2000), Difference Equations to Differential Equations, <http://math.furman.edu/~dcs/book/>
Strang, Gilbert (1991), Calculus, <http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm>
Stroyan, Keith D. (1997), A Brief Introduction to Infinitesimal Calculus, <http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm>
Wikibooks, Calculus, <http://en.wikibooks.org/wiki/Calculus>

External links

WIMS Function Calculator makes online calculation of derivatives; this software also enables interactive exercises. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Mathematics and Computer algebra, automatic differentiation, or AD sometimes alternatively called algorithmic differentiation, is a method to numerically In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, the differintegral is the combined differentiation / integration operator used in Fractional calculus. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics and its applications linearization refers to finding the Linear approximation to a function at a given point Numerical differentiation is a technique of Numerical analysis to produce an estimate of the Derivative of a Mathematical function or function Subroutine This article contains a list of techniques for the differentiation of real functions, categorized by type The primary operation in Differential calculus is finding a Derivative.
Online Derivatives Calculator.
Mathematical Assistant on Web online calculation of derivatives, including explanation of steps in the solution.
Proving Derivatives from First Principles.
Practice finding derivatives of randomly generated functions

Dictionary

-adjective

Imitative of the work of someone else.
(law, copyright) Referring to a work, such as a translation or adaptation, based on another work that may be subject to copyright restrictions.
(finance) Having a value that depends on an underlying asset of variable value.
Lacking originality.

-noun

Something derived.
(linguistics) A word that derives from another one.
(finance) A financial instrument whose value depends on the valuation of an underlying asset; such as a warrant, an option etc.
(chemistry) A chemical derived from another.
(calculus) The derived function of a function.
(calculus) The value of this function for a given value of its independent variable.

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