Thegraph of a function, drawn in black, and atangent line to that graph, drawn in red. Theslope of the tangent line is equal to the derivative of the function at the marked point.
The derivative at different points of a differentiable function. In this case, the derivative is equal to.
Inmathematics, thederivative is a fundamental tool that quantifies the sensitivity to change of afunction's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is theslope of thetangent line to thegraph of the function at that point. The tangent line is the bestlinear approximation of the function near that input value. For this reason, the derivative is often described as theinstantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable.[1] The process of finding a derivative is calleddifferentiation.
There are multiple differentnotations for differentiation.Leibniz notation, named afterGottfried Wilhelm Leibniz, is represented as the ratio of twodifferentials, whereasprime notation is written by adding aprime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. Thehigher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect totime is the object'svelocity, how the position changes as time advances, the second derivative is the object'sacceleration, how the velocity changes as time advances.
Derivatives can be generalized tofunctions of several real variables. In this case, the derivative is reinterpreted as alinear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. TheJacobian matrix is thematrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of thepartial derivatives with respect to the independent variables. For areal-valued function of several variables, the Jacobian matrix reduces to thegradient vector.
If the function is differentiable at, that is if the limit exists, then this limit is called thederivative of at. Multiple notations for the derivative exist.[4] The derivative of at can be denoted, read as " prime of"; or it can be denoted, read as "the derivative of with respect to at" or " by (or over) at". See§ Notation below. If is a function that has a derivative at every point in itsdomain, then a function can be defined by mapping every point to the value of the derivative of at. This function is written and is called thederivative function or thederivative of. The function sometimes has a derivative at most, but not all, points of its domain. The function whose value at equals whenever is defined and elsewhere is undefined is also called the derivative of. It is still a function, but its domain may be smaller than the domain of.[5]
For example, let be the squaring function:. Then the quotient in the definition of the derivative is[6]The division in the last step is valid as long as. The closer is to, the closer this expression becomes to the value. The limit exists, and for every input the limit is. So, the derivative of the squaring function is the doubling function:.
The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function, specifically the points and. As is made smaller, these points grow closer together, and the slope of this line approaches the limiting value, the slope of thetangent to the graph of at. In other words, the derivative is the slope of the tangent.[7]
Using infinitesimals
One way to think of the derivative is as the ratio of aninfinitesimal change in the output of the function to an infinitesimal change in its input.[8] In order to make this intuition rigorous, a system of rules for manipulating infinitesimal quantities is required.[9] The system ofhyperreal numbers is a way of treatinginfinite and infinitesimal quantities. The hyperreals are anextension of thereal numbers that contain numbers greater than anything of the form for any finite number of terms. Such numbers are infinite, and theirreciprocals are infinitesimals. The application of hyperreal numbers to the foundations of calculus is callednonstandard analysis. This provides a way to define the basic concepts of calculus such as the derivative and integral in terms of infinitesimals, thereby giving a precise meaning to the in the Leibniz notation. Thus, the derivative of becomes for an arbitrary infinitesimal, where denotes thestandard part function, which "rounds off" each finite hyperreal to the nearest real.[10] Taking the squaring function as an example again,
Continuity and differentiability
This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has ajump discontinuity).
The absolute value function is continuous but fails to be differentiable atx = 0 since the tangent slopes do not approach the same value from the left as they do from the right.
If isdifferentiable at, then must also becontinuous at.[11] As an example, choose a point and let be thestep function that returns the value 1 for all less than, and returns a different value 10 for all greater than or equal to. The function cannot have a derivative at. If is negative, then is on the low part of the step, so the secant line from to is very steep; as tends to zero, the slope tends to infinity. If is positive, then is on the high part of the step, so the secant line from to has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. However, even if a function is continuous at a point, it may not be differentiable there. For example, theabsolute value function given by is continuous at, but it is not differentiable there. If is positive, then the slope of the secant line from 0 to is one; if is negative, then the slope of the secant line from to is.[12] This can be seen graphically as a "kink" or a "cusp" in the graph at. Even a function with a smooth graph is not differentiable at a point where itstangent is vertical: For instance, the function given by is not differentiable at. In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.[13]
Most functions that occur in practice have derivatives at all points oralmost every point. Early in thehistory of calculus, many mathematicians assumed that a continuous function was differentiable at most points.[14] Under mild conditions (for example, if the function is amonotone or aLipschitz function), this is true. However, in 1872, Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as theWeierstrass function.[15] In 1931,Stefan Banach proved that the set of functions that have a derivative at some point is ameager set in the space of all continuous functions. Informally, this means that hardly any random continuous functions have a derivative at even one point.[16]
One common way of writing the derivative of a function isLeibniz notation, introduced byGottfried Wilhelm Leibniz in 1675, which denotes a derivative as the quotient of twodifferentials, such as and.[17] It is still commonly used when the equation is viewed as a functional relationship betweendependent and independent variables. The first derivative is denoted by, read as "the derivative of with respect to".[18] This derivative can alternately be treated as the application of adifferential operator to a function, Higher derivatives are expressed using the notation for the-th derivative of. These are abbreviations for multiple applications of the derivative operator; for example,[19] Unlike some alternatives, Leibniz notation involves explicit specification of the variable for differentiation, in the denominator, which removes ambiguity when working with multiple interrelated quantities. The derivative of acomposed function can be expressed using thechain rule: if and then[20]
Another common notation for differentiation is by using theprime mark in the symbol of a function. This notation, due toJoseph-Louis Lagrange, is now known asprime notation.[21] The first derivative is written as, read as " prime of", or, read as " prime".[22] Similarly, the second and the third derivatives can be written as and, respectively.[23] For denoting the number of higher derivatives beyond this point, some authors use Roman numerals insuperscript, whereas others place the number in parentheses, such as or.[24] The latter notation generalizes to yield the notation for theth derivative of.[19]
InNewton's notation or thedot notation, a dot is placed over a symbol to represent a time derivative. If is a function of, then the first and second derivatives can be written as and, respectively. This notation is used exclusively for derivatives with respect to time orarc length. It is typically used indifferential equations inphysics anddifferential geometry.[25] However, the dot notation becomes unmanageable for high-order derivatives (of order 4 or more) and cannot deal with multiple independent variables.
Another notation isD-notation, which represents the differential operator by the symbol.[19] The first derivative is written and higher derivatives are written with a superscript, so the-th derivative is. This notation is sometimes calledEuler notation, although it seems thatLeonhard Euler did not use it, and the notation was introduced byLouis François Antoine Arbogast.[26] To indicate a partial derivative, the variable differentiated by is indicated with a subscript, for example given the function, its partial derivative with respect to can be written or. Higher partial derivatives can be indicated by superscripts or multiple subscripts, e.g. and.[27]
In principle, the derivative of a function can be computed from the definition by considering the difference quotient and computing its limit. Once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed usingrules for obtaining derivatives of more complicated functions from simpler ones. This process of finding a derivative is known asdifferentiation.[28]
Given that the and are the functions. The following are some of the most basic rules for deducing the derivative of functions from derivatives of basic functions.[30]
The derivative of the function given by isHere the second term was computed using thechain rule and the third term using theproduct rule. The known derivatives of the elementary functions,,,, and, as well as the constant, were also used.
Anantiderivative of a function is a function whose derivative is. Antiderivatives are not unique: if is an antiderivative of, then so is, where is any constant, because the derivative of a constant is zero.[31] Thefundamental theorem of calculus shows that finding an antiderivative of a function gives a way to compute the areas of shapes bounded by that function. More precisely, theintegral of a function over aclosed interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of that interval.[32]
Higher-order derivatives
Higher order derivatives are the result of differentiating a function repeatedly. Given that is a differentiable function, the derivative of is the first derivative, denoted as. The derivative of is thesecond derivative, denoted as, and the derivative of is thethird derivative, denoted as. By continuing this process, if it exists, theth derivative is the derivative of theth derivative or thederivative of order. As has beendiscussed above, the generalization of derivative of a function may be denoted as.[33] A function that has successive derivatives is called times differentiable. If the-th derivative is continuous, then the function is said to be ofdifferentiability class.[34] A function that has infinitely many derivatives is calledinfinitely differentiable orsmooth.[35] Anypolynomial function is infinitely differentiable; taking derivatives repeatedly will eventually result in aconstant function, and all subsequent derivatives of that function are zero.[36]
Oneapplication of higher-order derivatives is inphysics. Suppose that a function represents the position of an object at the time. The first derivative of that function is thevelocity of an object with respect to time, the second derivative of the function is theacceleration of an object with respect to time,[28] and the third derivative is thejerk.[37]
Avector-valued function of a real variable sends real numbers to vectors in somevector space. A vector-valued function can be split up into its coordinate functions, meaning that. This includes, for example,parametric curves in or. The coordinate functions are real-valued functions, so the above definition of derivative applies to them. The derivative of is defined to be thevector, called thetangent vector, whose coordinates are the derivatives of the coordinate functions. That is,[38]if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of exists for every value of, then is another vector-valued function.[38]
Functions can depend uponmore than one variable. Apartial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used invector calculus anddifferential geometry. As with ordinary derivatives, multiple notations exist: the partial derivative of a function with respect to the variable is variously denoted by
,,,, or,
among other possibilities.[39] It can be thought of as the rate of change of the function in the-direction.[40] Here∂ is a roundedd called thepartial derivative symbol. To distinguish it from the letterd, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee".[41] For example, let, then the partial derivative of function with respect to both variables and are, respectively:In general, the partial derivative of a function in the direction at the point is defined to be:[42]
This is fundamental for the study of thefunctions of several real variables. Let be such areal-valued function. If all partial derivatives with respect to are defined at the point, these partial derivatives define the vectorwhich is called thegradient of at. If is differentiable at every point in some domain, then the gradient is avector-valued function that maps the point to the vector. Consequently, the gradient determines avector field.[43]
If is a real-valued function on, then the partial derivatives of measure its variation in the direction of the coordinate axes. For example, if is a function of and, then its partial derivatives measure the variation in in the and direction. However, they do not directly measure the variation of in any other direction, such as along the diagonal line. These are measured using directional derivatives. Given a vector, then thedirectional derivative of in the direction of at the point is:[44]
If all the partial derivatives of exist and are continuous at, then they determine the directional derivative of in the direction by the formula:[45]
When is a function from an open subset of to, then the directional derivative of in a chosen direction is the best linear approximation to at that point and in that direction. However, when, no single directional derivative can give a complete picture of the behavior of. The total derivative gives a complete picture by considering all directions at once. That is, for any vector starting at, the linear approximation formula holds:[46]Similarly with the single-variable derivative, is chosen so that the error in this approximation is as small as possible. The total derivative of at is the unique linear transformation such that[46]Here is a vector in, so the norm in the denominator is the standard length on. However, is a vector in, and the norm in the numerator is the standard length on.[46] If is a vector starting at, then is called thepushforward of by.[47]
If the total derivative exists at, then all the partial derivatives and directional derivatives of exist at, and for all, is the directional derivative of in the direction. If is written using coordinate functions, so that, then the total derivative can be expressed using the partial derivatives as amatrix. This matrix is called theJacobian matrix of at:[48]
The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as alinear approximation of the function at that point.
An important generalization of the derivative concernscomplex functions ofcomplex variables, such as functions from (a domain in) the complex numbers to. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition.[49] If is identified with by writing a complex number as then a differentiable function from to is certainly differentiable as a function from to (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 iscomplex linear and this imposes relations between the partial derivatives called theCauchy–Riemann equations – seeholomorphic functions.[50]
Another generalization concerns functions betweendifferentiable or smooth manifolds. Intuitively speaking such a manifold is a space that can be approximated near each point by a vector space called itstangent space: the prototypical example is asmooth surface in. The derivative (or differential) of a (differentiable) map between manifolds, at a point in, is then alinear map from the tangent space of at to the tangent space of at. The derivative function becomes a map between thetangent bundles of and. This definition is used indifferential geometry.[51]
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 allcontinuous functions and many other functions can be differentiated using a concept known as theweak derivative. The idea is to embed the continuous functions in a larger space called the space ofdistributions and only require that a function is differentiable "on average".[53]
Properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology; an example isdifferential algebra. Here, it consists of the derivation of some topics in abstract algebra, such asrings,ideals,field, and so on.[54]
The discrete equivalent of differentiation isfinite differences. The study of differential calculus is unified with the calculus of finite differences intime scale calculus.[55]
^In the formulation of calculus in terms of limits, various authors have assigned the symbol various meanings. Some authors such asVarberg, Purcell & Rigdon 2007, p. 119 andStewart 2002, p. 177 do not assign a meaning to by itself, but only as part of the symbol. Others define as an independent variable, and define by.Innon-standard analysis is defined as an infinitesimal. It is also interpreted as theexterior derivative of a function. Seedifferential (infinitesimal) for further information.
^Varberg, Purcell & Rigdon 2007. See p. 133 for the power rule, pp. 115–116 for the trigonometric functions, p. 326 for the natural logarithm, pp. 338–339 for exponential with base, p. 343 for the exponential with base, p. 344 for the logarithm with base, and p. 369 for the inverse of trigonometric functions.
^ For constant rule and sum rule, seeApostol 1967, pp. 161, 164, respectively. For the product rule, quotient rule, and chain rule, seeVarberg, Purcell & Rigdon 2007, pp. 111–112, 119, respectively. For the special case of the product rule, that is, the product of a constant and a function, seeVarberg, Purcell & Rigdon 2007, pp. 108–109.
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Larson, Ron; Hostetler, Robert P.; Edwards, Bruce H. (February 28, 2006),Calculus: Early Transcendental Functions (4th ed.), Houghton Mifflin Company,ISBN978-0-618-60624-5
