Derivative
Part of a series of articles about |
Calculus |
---|
The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable.[1] The process of finding a derivative is called differentiation.
There are multiple different notations for differentiation, two of the most commonly used being
Derivatives can be generalized to
Definition
As a limit
A function of a real variable is differentiable at a point of its
If the function is differentiable at , that is if the limit exists, then this limit is called the derivative 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 its domain, 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 the derivative function or the derivative 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 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 the tangent 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 an infinitesimal 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 of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals are an extension of the real numbers that contain numbers greater than anything of the form for any finite number of terms. Such numbers are infinite, and their reciprocals are infinitesimals. The application of hyperreal numbers to the foundations of calculus is called nonstandard analysis. This provides a way to define the basic concepts of calculus such as the derivative and integral in terms of infinitesimals, thereby giving a precise meaning to the in the Leibniz notation. Thus, the derivative of becomes
Continuity and differentiability
If is
Most functions that occur in practice have derivatives at all points or
Notation
One common symbol for the derivative of a function is
Another common notation for differentiation is by using the prime mark in the symbol of a function . This is known as prime notation, due to the Joseph-Louis Lagrange.[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 in superscript, whereas others place the number in parentheses, such as or [24] The latter notation generalizes to yield the notation for the -th derivative of .[19]
In
Another notation is D-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 called Euler notation, although it seems that Leonhard Euler did not use it, and the notation was introduced by Louis 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]
Rules of computation
In principle, the derivative of a function can be computed from the definition by considering the difference quotient and computing its limit. Once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones. This process of finding a derivative is known as differentiation.[28]
Rules for basic functions
The following are the rules for the derivatives of the most common basic functions. Here, is a real number, and is the mathematical constant approximately 2.71828.[29]
- Derivatives of powers:
- Functions of exponential, natural logarithm, and logarithm with general base:
- , for
- , for
- , for
- Trigonometric functions:
- Inverse trigonometric functions:
- , for
- , for
Rules for combined functions
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]
- Constant rule: if is constant, then for all ,
- Sum rule:
- for all functions and and all real numbers and .
- Product rule:
- for all functions and . As a special case, this rule includes the fact whenever is a constant because by the constant rule.
- Quotient rule:
- for all functions and at all inputs where g ≠ 0.
- Chain rule for composite functions: If , then
Computation example
The derivative of the function given by is
Higher-order derivatives
Higher order derivatives means that a function is differentiated repeatedly. Given that is a differentiable function, the derivative of is the first derivative, denoted as . The derivative of is the second derivative, denoted as , and the derivative of is the third derivative, denoted as . By continuing this process, if it exists, the -th derivative as the derivative of the -th derivative or the derivative of order . As has been discussed above, the generalization of derivative of a function may be denoted as .[31] A function that has successive derivatives is called times differentiable. If the -th derivative is continuous, then the function is said to be of
In one of its applications, the higher-order derivatives may have specific interpretations in physics. Suppose that a function represents the position of an object at the time. The first derivative of that function is the velocity of an object with respect to time, the second derivative of the function is the acceleration of an object with respect to time,[28] and the third derivative is the jerk.[35]
In other dimensions
Vector-valued functions
A vector-valued function of a real variable sends real numbers to vectors in some vector space . A vector-valued function can be split up into its coordinate functions , meaning that . This includes, for example,
Partial derivatives
Functions can depend upon more than one variable. A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in vector calculus and differential geometry. As with ordinary derivatives, multiple notations exist: the partial derivative of a function with respect to the variable is variously denoted by
among other possibilities.[37] It can be thought of as the rate of change of the function in the -direction.[38] 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".[39] For example, let , then the partial derivative of function with respect to both variables and are, respectively:
This is fundamental for the study of the
Directional derivatives
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. Choose a vector , then the directional derivative of in the direction of at the point is:[42]
If all the partial derivatives of exist and are continuous at , then they determine the directional derivative of in the direction by the formula:[43]
Total derivative, total differential and Jacobian matrix
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:[44]
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 a
Generalizations
The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a linear approximation of the function at that point.
- An important generalization of the derivative concerns complex functions of complex variables, such as functions from (a domain in) the complex numbers to . The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition.[47] 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 is complex linear and this imposes relations between the partial derivatives called the Cauchy–Riemann equations – see holomorphic functions.[48]
- Another generalization concerns functions between differentiable or smooth manifolds. Intuitively speaking such a manifold is a space that can be approximated near each point by a vector space called itssmooth surfacein . The derivative (or differential) of a (differentiable) map between manifolds, at a point in , is then a linear map from the tangent space of at to the tangent space of at . The derivative function becomes a map between the tangent bundles of and . This definition is used in differential geometry.[49]
- Differentiation can also be defined for maps between vector space, such as Banach space, in which those generalizations are the Gateaux derivative and the Fréchet derivative.[50]
- 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".[51]
- Properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology; an example is differential algebra. Here, it consists of the derivation of some topics in abstract algebra, such as rings, ideals, field, and so on.[52]
- The discrete equivalent of differentiation is time scale calculus.[53]
- The prime factorization. This is an analogy with the product rule.[54]
See also
Notes
- ^ Stewart 2002, p. 129–130.
- ^ Stewart 2002, p. 127; Strang et al. 2023, p. 220.
- ^ Gonick 2012, p. 83.
- ^ Gonick 2012, p. 88; Strang et al. 2023, p. 234.
- ^ Gonick 2012, p. 83; Strang et al. 2023, p. 232.
- ^ Gonick 2012, pp. 77–80.
- ^ Thompson 1998, pp. 34, 104; Stewart 2002, p. 128.
- ^ Thompson 1998, pp. 84–85.
- ^ Keisler 2012, pp. 902–904.
- ^ Keisler 2012, p. 45; Henle & Kleinberg 2003, p. 66.
- ^ a b Gonick 2012, p. 156.
- ^ Gonick 2012, p. 149.
- ^ Jašek 1922; Jarník 1922; Rychlík 1923.
- ^ David 2018.
- ^ Banach 1931, cited in Hewitt & Stromberg 1965.
- ^ Apostol 1967, p. 172.
- ^ Cajori 2007, p. 204.
- ^ Moore & Siegel 2013, p. 110.
- ^ a b c Varberg, Purcell & Rigdon 2007, p. 125–126.
- ^ In the formulation of calculus in terms of limits, various authors have assigned the symbol various meanings. Some authors such as Varberg, Purcell & Rigdon 2007, p. 119 and Stewart 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 analysisis defined as an infinitesimal. It is also interpreted as the exterior derivative of a function . Seedifferential (infinitesimal)for further information.
- ^ Schwartzman 1994, p. 171.
- ^ Moore & Siegel 2013, p. 110; Goodman 1963, p. 78–79.
- ^ Varberg, Purcell & Rigdon 2007, p. 125–126; Cajori 2007, p. 228.
- ^ Choudary & Niculescu 2014, p. 222; Apostol 1967, p. 171.
- ^ Evans 1999, p. 63; Kreyszig 1991, p. 1.
- ^ Cajori 1923.
- ^ Apostol 1967, p. 172; Varberg, Purcell & Rigdon 2007, p. 125–126.
- ^ a b Apostol 1967, p. 160.
- ^ Varberg, Purcell & Rigdon 2007. See p. 133 for the power rule, p. 115–116 for the trigonometric functions, p. 326 for the natural logarithm, p. 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, see Apostol 1967, p. 161, 164, respectively. For the product rule, quotient rule, and chain rule, see Varberg, Purcell & Rigdon 2007, p. 111–112, 119, respectively. For the special case of the product rule, that is, the product of a constant and a function, see Varberg, Purcell & Rigdon 2007, p. 108–109.
- ^ Apostol 1967, p. 160; Varberg, Purcell & Rigdon 2007, p. 125–126.
- ^ Warner 1983, p. 5.
- ^ Debnath & Shah 2015, p. 40.
- ^ Carothers 2000, p. 176.
- ^ Stewart 2002, p. 193.
- ^ a b Stewart 2002, p. 893.
- ^ Stewart 2002, p. 947; Christopher 2013, p. 682.
- ^ Stewart 2002, p. 949.
- ^ Silverman 1989, p. 216; Bhardwaj 2005, See p. 6.4.
- ^ Mathai & Haubold 2017, p. 52.
- ^ Gbur 2011, pp. 36–37.
- ^ Varberg, Purcell & Rigdon 2007, p. 642.
- ^ Guzman 2003, p. 35.
- ^ a b c Davvaz 2023, p. 266.
- ^ Lee 2013, p. 72.
- ^ Davvaz 2023, p. 267.
- ^ Roussos 2014, p. 303.
- ^ Gbur 2011, pp. 261–264.
- ^ Gray, Abbena & Salamon 2006, p. 826.
- ^ Azegami 2020. See p. 209 for the Gateaux derivative, and p. 211 for the Fréchet derivative.
- ^ Funaro 1992, p. 84–85.
- ^ Kolchin 1973, p. 58, 126.
- ^ Georgiev 2018, p. 8.
- ^ Barbeau 1961.
References
- ISBN 978-0-471-00005-1
- Azegami, Hideyuki (2020), Shape Optimization Problems, Springer Optimization and Its Applications, vol. 164, Springer, S2CID 226442409
- .
- Barbeau, E. J. (1961). "Remarks on an arithmetic derivative". Zbl 0101.03702.
- Bhardwaj, R. S. (2005), Mathematics for Economics & Business (2nd ed.), Excel Books India, ISBN 9788174464507
- JSTOR 1967725
- Cajori, Florian (2007), A History of Mathematical Notations, vol. 2, Cosimo Classics, ISBN 978-1-60206-713-4
- Carothers, N. L. (2000), Real Analysis, Cambridge University Press
- Choudary, A. D. R.; Niculescu, Constantin P. (2014), Real Analysis on Intervals, Springer India, ISBN 978-81-322-2148-7
- Christopher, Essex (2013), Calculus: A complete course, Pearson, p. 682, OCLC 872345701
- ISBN 978-3-540-65058-4
- David, Claire (2018), "Bypassing dynamical systems: A simple way to get the box-counting dimension of the graph of the Weierstrass function", Proceedings of the International Geometry Center, 11 (2), Academy of Sciences of Ukraine: 53–68,
- Davvaz, Bijan (2023), Vectors and Functions of Several Variables, Springer, S2CID 259885793
- Debnath, Lokenath; Shah, Firdous Ahmad (2015), Wavelet Transforms and Their Applications (2nd ed.), Birkhäuser, ISBN 978-0-8176-8418-1
- ISBN 0-8218-0772-2
- Eves, Howard (January 2, 1990), An Introduction to the History of Mathematics (6th ed.), Brooks Cole, ISBN 978-0-03-029558-4
- Funaro, Daniele (1992), Polynomial Approximation of Differential Equations, Lecture Notes in Physics Monographs, vol. 8, Springer, ISBN 978-3-540-46783-0
- ISBN 978-1-139-49269-0
- Georgiev, Svetlin G. (2018), Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales, Springer, ISBN 978-3-319-73954-0
- Goodman, A. W. (1963), Analytic Geometry and the Calculus, The MacMillan Company
- ISBN 978-0-06-168909-3
- Gray, Alfred; Abbena, Elsa; Salamon, Simon (2006), Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC Press, ISBN 978-1-58488-448-4
- Guzman, Alberto (2003), Derivatives and Integrals of Multivariable Functions, Springer, ISBN 978-1-4612-0035-2
- Henle, James M.; Kleinberg, Eugene M. (2003), Infinitesimal Calculus, Dover Publications, ISBN 978-0-486-42886-4
- ISBN 978-3-662-28275-5
- Jašek, Martin (1922), "Funkce Bolzanova" (PDF), Časopis pro Pěstování Matematiky a Fyziky (in Czech), 51 (2): 69–76,
- doi:10.21136/CPMF.1922.109021. See the English version here.
- Keisler, H. Jerome (2012) [1986], Elementary Calculus: An Approach Using Infinitesimals (2nd ed.), Prindle, Weber & Schmidt, ISBN 978-0-871-50911-6
- Kolchin, Ellis (1973), Differential Algebra And Algebraic Groups, Academic Press, ISBN 978-0-08-087369-5
- Kreyszig, Erwin (1991), Differential Geometry, New York: ISBN 0-486-66721-9
- Larson, Ron; Hostetler, Robert P.; Edwards, Bruce H. (February 28, 2006), Calculus: Early Transcendental Functions (4th ed.), Houghton Mifflin Company, ISBN 978-0-618-60624-5
- Lee, John M. (2013), Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, Springer, ISBN 978-0-387-21752-9
- Mathai, A. M.; Haubold, H. J. (2017), Fractional and Multivariable Calculus: Model Building and Optimization Problems, Springer, ISBN 978-3-319-59993-9
- Moore, Will H.; Siegel, David A. (2013), A Mathematical Course for Political and Social Research, Princeton University Press, ISBN 978-0-691-15995-9
- Roussos, Ioannis M. (2014), Improper Riemann Integral, ISBN 978-1-4665-8807-3
- Rychlík, Karel (1923), Über eine Funktion aus Bolzanos handschriftlichem Nachlasse
- Schwartzman, Steven (1994), The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, Mathematical Association of American, ISBN 9781614445012
- Silverman, Richard A. (1989), Essential Calculus: With Applications, Courier Corporation, ISBN 9780486660974
- ISBN 978-0-534-39339-7
- ISBN 978-1-947172-13-5
- ISBN 978-0-312-18548-0
- Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), Calculus (9th ed.), ISBN 978-0131469686
- ISBN 978-0-387-90894-6
External links
- "Derivative", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Khan Academy: "Newton, Leibniz, and Usain Bolt"
- Weisstein, Eric W. "Derivative". MathWorld.
- Online Derivative Calculator from Wolfram Alpha.