Multiple issues refimprove September 2009 cleanup September 2009 essay September 2009 An energyderivative is a derivative contract based on derived from an underlying energy asset, such as natural gas, crude oil, or electricity ref Investopedia article on energy derivatives ref . Energy derivatives include exchange traded contracts such as Futures contract futures and Option finance options , and over the counter privately negotiated derivatives such as forwards, Swap finance swaps and options. Major players in the energyderivative markets include major trading houses, oil companies, utilities, financial institutions. Energy derivatives have received criticism after the 2008 financial collapse, critics cite evidence that the market artificially inflates the price of oil and other energy providers. ref cite web title Behind Oil Price Rise Peak Oil or Wall Street Speculation? url http axisoflogic.com artman publish Article 64370.shtml publisher Axis of Logic accessdate 29 March 2012 author F. William Engdahl date Mar 18, 2012 ref See Exotic derivatives . Definition The basic building blocks for all derivative contracts are the Futures and swaps contracts. For the energy market s these are traded in New York NYMEX , in Tokyo TOCOM and on line through IntercontinentalExchange . A future ... Petroleum Exchange IPE . Applications There are 3 principal applications for the energyderivative ... Reflist External links http www.investopedia.com terms e energy derivative.asp Investopedia article on energy derivatives DEFAULTSORT EnergyDerivative Category Derivatives finance Category Energy ... with other party considering all these parameters. History The first energy derivatives covered ... in the 1970s. At roughly the same time, energy products began trading on derivatives exchange with crude .... Any subsequent rise in the jet price for the period is protected by the derivative transaction ... to be considered when using energy derivatives to manage risk. A key consideration ... more details
multiple issues cleanup August 2009 refimprove August 2009 A renewable energyderivative is based on a new method of securitization, which is a structured finance process that Resource pooling pools and repackaging repackages cash flow producing financial assets into securities which are then sold to investors. The term securitization is derived from the fact that securities are used to obtain funds from investors. ref http www.awea.org greenpower gp how2.html What Are Tradable Renewable Certificates? , Green Power, American Wind Energy Association. ref Summary Cash flow producing financial assets were traditionally mortgages , credit cards, and then student loans. More recently the assets have become more exotic, like annual airline flight tax , intellectual property such as Bowie Bonds , and life insurance premiums for captive killer whale orcas . In essence any asset s cash flow can be securitized if the historical, realized cash flow demonstrates a statistical predictability. This process was first applied to renewable energy by Joseph Brant Arseneau and his team at IBM . A renewable energyderivative is just that a cash flow that has been proven historically and demonstrates a statistical predictability. The predictability of the cash flow is achieved by consulting with professionals to ensure that the primary asset, in this case a private home, is equipped with the optimal types of renewable energy sources for its immediate environment. These energy sources are chosen with respect to the location of the home, the weather, and supporting data, including historical wind patterns, heat changes, and earth core temperatures. If the assets produce more power than is required by the home, the excess power is sold to a Special Purpose Vehicle SVP , which pools all of the surrounding ... to the capital markets as securities. History Energy securitization began in 2006. Pre Finance Activity Constructing Renewable Energy Housing Empty section date January 2011 Structure Pooling and transfer ... more details
, see Differential calculus . For other uses, see Derivative disambiguation . Good Article ..., drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point. Calculus cTopic Differentiation In calculus , a branch of mathematics , the derivative ... speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity for example, the derivative of the position of a moving object with respect to time is the object s instantaneous velocity . The derivative of a function at a chosen input value ... of a single real variable, the derivative at a point equals the slope of the Tangent tangent line to the graph of a function graph of the function at that point. In higher dimensions, the derivative ... of finding a derivative is called differentiation . The reverse process is called antiderivative ... in single variable calculus. Differentiation and the derivative File Graph of sliding derivative line.gif left thumb 400px At each point, the derivative of math f x 1 x sin x 2 math is the slope ... its slope is the derivative. Note derivative is positive number positive where green, negative number ... of change is called the derivative of y with respect to x . In more precise language, the dependence ... of a function graph of y is plotted against x , the derivative measures the slope of this graph ... small. In Leibniz s notation , such an infinitesimal change in x is denoted by dx , and the derivative ... quantities. The above expression is read as the derivative of y with respect to x , d y by d .... The derivative of y with respect to x at a is, geometrically, the slope of the tangent line ... x h . math This expression is Isaac Newton Newton s difference quotient . The derivative is the value of the difference quotient as the secant lines approach the tangent line. Formally, the derivative ... function differentiable at a . Here f &prime a is one of several common notations for the derivative ... more details
force is the time derivative of momentum Power physics power is the time derivative of energy electrical current is the time derivative of electric charge and so on. A common occurrence in physics is the time derivative of a vector geometric vector , such as velocity or displacement. In dealing with such a derivative ...A time derivative is a derivative of a function with respect to time , usually interpreted as the Derivative ... derivative. In addition to the normal Derivative Leibniz s notation Leibniz s notation, math frac ... dot x math Derivative Newton s notation Newton s notation , and adding a Prime symbol prime to the function, math x t , math Derivative Lagrange s notation Lagrange s notation . These two shorthands ... derivative with respect to time is written as math frac d 2x dt 2 math with the corresponding shorthands of math ddot x math and math x t math . As a generalization, the time derivative of a vector ... derivative math dot x math is its velocity , and its second derivative with respect to time, math ddot x math , is its acceleration . Even higher derivatives are sometimes also used the third derivative ..., the velocity now is found. The time derivative of the displacement vector is the velocity vector. In general, the derivative of a vector is a vector made up of components each of which is the derivative of the corresponding component of the original vector. Thus, in this case, the velocity ... derivative of velocity math mathbf a t frac d , mathbf v t dt x t , y t mathbf r t , . math The acceleration ... time derivative, a stocks and flows flow variable . Examples include The flow of net fixed investment is the time derivative of the capital stock . The flow of inventory investment is the time derivative of the stock of inventories . The growth rate of the money supply is the time derivative of the money supply divided by the money supply itself. Sometimes the time derivative of a flow variable can appear in a model The growth rate of output is the time derivative of the flow of output divided ... more details
wiktionarypar derivativeDerivative , in calculus, is a measurement of how a function changes when the values of its inputs change. Derivative may also refer to Derivative finance , a contract whose value is derived from that of other quantities Derivative chemistry , a type of compound which is a product of the process of derivatization A derivative genre in music , which usually starts as a subgenre and becomes mainstream after the original genre declines in popularity. Example house music is a derivative of disco . Derivative Inc., a spin off of Side Effects Software, creators of Houdini software Derivative suit or derivative action, a type of lawsuit filed by shareholders of a corporation Derivative work , in copyright law, a modification of an original work Formal derivative , in mathematics, an operation on elements of a polynomial ring which mimics the form of the derivative from calculus Normal function Properties Derivative of a normal function in set theory Aeroderivative gas turbine , a mechanical drive gas turbine derived from an aero engine gas turbine Derivative linguistics See also Derivation disambiguation Derived , in phylogenetics, a trait present in an organism, but absent in the last common ancestor of its group Disambig bg cs Deriv t da Derivat de Derivat eo Deriva o fr D riv homonymie is Aflei a he lb Derivat nl Derivaat pl Pochodna ujednoznacznienie ru sk Deriv t sr Derivat th Derivative tr T rev anlam ayr m ... more details
In mathematics In abstract algebra and mathematical logic a derivative algebra abstract algebra derivative algebra is an algebraic structure that provides an abstraction of the derivative operator in topological space topology and which provides algebraic semantics for the modal logic wK3 . In differential geometry a derivative algebra is a vector space with a product operation that has similar behaviour to the standard cross product of 3 vector geometric vector s. Citation needed date July 2009 disambig ... more details
derivative deal was in July 1996 when Aquila Energy structured a dual commodity hedge for Consolidated ... derivative contract because if it rains the day of the sporting event, fewer tickets will be sold. Heating degree days are one of the most common types of weather derivative. Typical terms for an HDD ... options , in 1999. The CME currently trades weather derivative contracts for 18 cities in the United .... That is due to the fact that underlying asset of the weather derivative is non tradeable which violates ... of ways Business pricing Business pricing requires the company utilizing weather derivative instruments ... payout of the derivative is computed to find the expectation. The method is very quick and simple ... the derivative value is determined for example monthly seasonal cummulative heating degree day s . The simplest .... Ensemble forecasting Ensemble forecasts are especially appropriate for weather derivative pricing within the contract period of a monthly temperature derivative. However, individuals members of the ensemble ... to weather der.pdf Introduction to Weather Derivatives , Weather Derivatives Group, Aquila Energy. Peter ... weather forecast 2008 06 09 weather derivative N.htm Weather Derivatives becoming hot ..., R. S., Ed. 2002 . http riskbooks.com Energy 20and 20Commodities Climate Risk Weather Market.php ... for pricing weather derivatives and managing a portfolio of weather derivative risk http www.speedwellweather.com ... www.yjenergy.com YellowJacket OTC Weather Derivative Trading and Network http www.cmegroup.com trading ... more details
derivative of the Coulomb potential energy functional is math frac delta 2 J rho delta rho mathbf ...Merge from First variation discuss Talk Functional derivative Merge proposal date July 2009 In mathematics and theoretical physics , the functional derivative is a generalization of the directional derivative ... one dimensional derivative in usual calculus . The mathematically formal treatment is the subject ... M rightarrow mathbb C math , the functional derivative of F , denoted math delta F delta varphi ... derivative along that function. In physics, it s common to use the Dirac delta function math delta x y math in place of a generic test function math f x math , for yielding the functional derivative at the point math y math this is a point of the whole functional derivative as a partial derivative ... derivative may be made more mathematically precise and formal by defining the topological vector space ... derivative becomes known as the Fr chet derivative , while one uses the G teaux derivative ... derivative describes how the functional math F varphi x math changes as a result of a small change ... of math rho r math . Applying the definition of functional derivative, math begin align left langle ... frac delta V r delta rho r frac 1 4 pi epsilon 0 r r . math Now we can evaluate the functional derivative ... equation indeed, the functional derivative was introduced in physics within the derivation of the Joseph ... of math mathbf r math , the scalar product of the functional derivative with a function math phi ... at the integration boundaries. Thus the functional derivative is math frac delta F rho delta rho frac ... n i math components math mathbf r in mathbb R n math are all partial derivative operators of order ... . end align math Thomas Fermi kinetic energy functional The Thomas Fermi model of 1927 used a kinetic energy functional for a noninteracting uniform free electron model electron gas in a first attempt ... potential energy functional For the classical part of the potential, Thomas and Fermi employed the Coulomb ... more details
In mathematics , the quasi derivative is one of several generalizations of the derivative of a function mathematics function between two Banach space s. The quasi derivative is a slightly stronger version of the G teaux derivative , though weaker than the Fr chet derivative . Let f A &rarr F be a continuous function from an open set A in a Banach space E to another Banach space F . Then the quasi derivative of f at x sub 0 sub &isin A is a linear transformation u E &rarr F with the following property for every continuous function g 0,1 &rarr A with g 0 x sub 0 sub such that g &prime 0 &isin E exists, math lim t to 0 frac f g t f x 0 t u g 0 . math If such a linear map u exists, then f is said to be quasi differentiable at x sub 0 sub . Continuity of u need not be assumed, but it follows instead from the definition of the quasi derivative. If f is Fr chet differentiable at x sub 0 sub , then by the chain rule , f is also quasi differentiable and its quasi derivative is equal to its Fr chet derivative at x sub 0 sub . The converse is true provided E is finite dimensional. Finally, if f is quasi differentiable, then it is G teaux differentiable and its G teaux derivative is equal to its quasi derivative. References cite book author Dieudonn , J title Foundations of modern analysis publisher Academic Press year 1969 math stub Category Banach spaces Category Generalizations of the derivative pl Quasi pochodna ... more details
Unreferenced date December 2009 In calculus , the derivative of a constant function is 0 number zero A constant function is one that does not depend on the independent variable, such as f x 7 . The rule can be justified in various ways. The derivative is the slope of the tangent to the given function s graph, and the graph of a constant function is a horizontal line, whose slope is zero. Proof A formal proof , from the Derivative Definition via difference quotients definition of a derivative , is math f x lim h to 0 frac f x h f x h lim h to 0 frac c c h lim h to 0 0 0. math In Leibniz notation , it is written as math frac d dx c 0. math Antiderivative of zero A partial converse to this statement is the following If a function has a derivative of zero on an interval, it must be constant on that interval. This is not a consequence of the original statement, but follows from the mean value theorem . It can be generalized to the statement that If two functions have the same derivative on an interval, they must differ by a constant, or If g is an antiderivative of f on and interval, then all antiderivatives of &fnof on that interval are of the form g x C, where C is a constant. From this follows a weak version of the second fundamental theorem of calculus if is continuous on a,b and g for some function g, then math int a b f x , dx g b g a . math DEFAULTSORT Derivative Of A Constant Category Differential calculus Category Zero ca Derivada d una constant ... more details
In mathematics , the symmetric derivative is an Operator mathematics operation related to the ordinary derivative . It is defined as math lim h to 0 frac f x h f x h 2h . math A function is symmetrically differentiable at a point x if its symmetric derivative exists at that point. It can be shown that if a function is differentiable function differentiable at a point, it is also symmetrically differentiable, but the converse is not true. The best known example is the absolute value function f x x , which is not differentiable at x 0, but is symmetrically differentiable here with symmetric derivative 0. It can also be shown that the symmetric derivative at a point is the mean of the one sided derivatives at that point, if they both exist. See also Symmetrically continuous function References cite book first Brian S. last Thomson year 1994 title Symmetric Properties of Real Functions publisher Marcel Dekker isbn 0 8247 9230 0 Category Differential calculus mathanalysis stub bs Simetri na derivacija ca Derivada sim trica eo Simetria deriva o pt Derivada sim trica ... more details
derivative of a function mathematics function f is defined by the formula math frac f f math where f &prime is the derivative of f . When f is a function f x of a real variable x , and takes real numbers real , strictly Positive number positive values, this is equal to the derivative of ln f or, the derivative of the natural logarithm of f . This follows directly from the chain rule . Basic properties Many properties of the real logarithm also apply to the logarithmic derivative, even ... So for positive real valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. But we can also use the Leibniz law for the derivative of a product ... that the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors when they are defined . Similarly in fact this is a consequence , the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function math ... number is the negation of the logarithm of the number. More generally, the logarithmic derivative ... derivative of a power with constant real exponent is the product of the exponent and the logarithmic derivative of the base math frac u k u k frac ku k 1 u u k k frac u u , math just as the logarithm ... derivative. Computing ordinary derivatives using logarithmic derivatives Main Logarithmic differentiation ... to compute Nowrap &fnof x . Instead of computing it directly, we compute its logarithmic derivative ... the logarithmic derivative of each factor, summing, and multiplying by . Integrating factors The logarithmic derivative idea is closely connected to the integrating factor method for first order ... rule as D M where M now denotes the multiplication operator by the logarithmic derivative ... derivative behaves in a way that is easily analysed in terms of the particular case z sup n sup with n an integer, n 0. The logarithmic derivative is then n z and one can draw the general ... more details
The derivative is a fundamental construction of differential calculus and admits many possible generalizations ... case is the functional derivative variational derivative in the calculus of variations . Repeated ... calculus main Fr chet derivative The derivative is often met for the first time as an operation ... x h math exists. Its value is then the derivative x . A function is differentiable on an Interval ... L z f x z f x x f x math is tangent to the original function at the point math x, f x math , the derivative ... case coincides with the original definition. In this case the derivative is represented by a 1 by 1 ... of this matrix represents a partial derivative , specifying the rate of change of one range coordinate ... from R sup n sup to R scalar field s , the total derivative can be interpreted as a vector field ... directional derivative s of Scalar mathematics scalar functions or normal directions. Several linear ... valued functions from R to R sup n sup i.e., parametric curve s , one can take the derivative of each component separately. The resulting derivative is another vector valued function. This is useful, for example, if the vector valued function is the position vector of a particle through time, then the derivative is the velocity vector of the particle through time. The convective derivative takes into account ... The subderivative and subgradient are generalizations of the derivative to convex function s. Higher ... of repeatedly applying the derivative, one repeatedly applies partial derivative s with respect to different ... orders, which are studied in fractional calculus . The 1 order derivative corresponds to the integral ... a Fr chet derivative suitably extended definition of differentiability . The Schwarzian derivative ... way that a normal derivative describes how a function is approximated by a linear map. Functional analysis In functional analysis , the functional derivative defines the derivative with respect to a function of a functional on a space of functions. This is an extension of the directional derivative ... more details
Orphan date February 2009 A derivative chromosome der is a structurally rearranged chromosome generated either by a rearrangement involving two or more chromosomes or by multiple aberrations within a single chromosome e.g. an inversion and a deletion of the same chromosome, or deletions in both arms of a single chromosome . http ghr.nlm.nih.gov ghr glossary derivativechromosome The term always refers to the chromosome that has an intact centromere. Derivative chromosomes are designated by the abbreviation der when used to describe a Karyotype . The derivative chromosome must be specified in parentheses followed by all aberrations involved in this derivative chromosome. The aberrations must be listed from pter to qter and not be separated by a comma. For example, 46,XY,der 4 t 4 8 p16 q22 t 4 9 q31 q31 would refer to a derivative chromosome 4 which is the result of a translocation between the short arm of chromosome 4 at band 16 and the long arm of chromosome 8 at band 22, and a translocation between the long arm of chromosome 4 at band 31 and the long arm of chromosome 9 at band 31. Derivative chromosomes and other abnormalities could be drawn online using CyDAS online tools Hiller B, Bradtke J, Balz H and Rieder H 2004 CyDAS Online Analysis Site , http www.cydas.org OnlineAnalysis References An International System for Human Cytogenetic Nomenclature, Shaffer, L.G., Tommerup N. eds S. Karger, Basel 2005 Category Chromosomes genetics stub ... more details
Orphan date September 2011 In mathematics, the Pansu derivative is a derivative on a Carnot group , introduced by harvtxt Pansu 1989 . References Citation last1 Pansu first1 Pierre title M triques de Carnot Carath odory et quasiisom tries des espaces sym triques de rang un doi 10.2307 1971484 mr 979599 year 1989 journal Annals of Mathematics Annals of Mathematics. Second Series issn 0003 486X volume 129 issue 1 pages 1 60 Category Lie groups ... more details
Orphan date February 2009 Derivative Dribble is a blog written by Charles Davi focused on finance , particularly Derivative finance derivatives and structured products . The site explains how various financial instruments work and why they are used. Additionally, the site has many opinion pieces on how derivatives and structured products operate in the broader financial system . Recognition The site is widely referenced in leading finance blogs ref http ftalphaville.ft.com blog 2008 12 08 50142 further reading 170 ref ref http acrossthecurve.com ?s Derivative Dribble ref ref http paul.kedrosky.com archives 2008 12 09 afternoon readi.html ref and is syndicated on the Atlantic Monthly s Business Channel ref http business.theatlantic.com author charles davi ref and Nouriel Roubini s RGE Monitor ref http www.rgemonitor.com globalmacro monitor author name cdavi3 Charles Davi ref . References Reflist External links http www.derivativedribble.wordpress.com Derivative Dribble Category Blogs ... more details
math is defined to be the adjoint of the Malliavin derivative math delta left mathrm D t right operatorname ... R . math See also H derivative References reflist Category Generalizations of the derivative Category ... more details
In the context of Lagrangian Mechanics the fiber derivative is used to convert between the Lagrangian and Hamiltonian Mechanics Hamiltonian forms. In particular, if math Q math is the configuration manifold then the Lagrangian math L math is defined on the tangent bundle math TQ math and the Hamiltonian is defined on the cotangent bundle math T Q math the fiber derivative is a map math mathbb F L TQ rightarrow T Q math such that math mathbb F L v cdot w frac d ds s 0 L v sw math , where math v math and math w math are vectors from the same tangent space. When restricted to a particular point, the fiber derivative is a Legendre transformation . Category Lagrangian mechanics math stub physics stub ... more details
distinguish Differentiation in Fr chet spaces In mathematics , the Fr chet derivative is a derivative defined on Banach space s. Named after Maurice Fr chet , it is commonly used to generalize the derivative ... real variables, and to define the functional derivative used widely in the calculus of variations . Generally, it extends the idea of the derivative from real valued function mathematics function s of one real variable to functions on Banach spaces. The Fr chet derivative should be contrasted to the more general G teaux derivative which is a generalization of the classical directional derivative . The Fr chet derivative has applications to nonlinear problems throughout mathematical analysis ... exists, we write math Df x A x math and call it the Fr chet derivative of f at x . A function f that is Fr chet differentiable for any point of U , and whose derivative Df x is continuous in x on U , is said to be C sup 1 sup . This notion of derivative is a generalization of the ordinary derivative ... W is differentiable at y f x , then the composition g o f is differentiable in x and the derivative ... derivative in finite dimensional spaces is the usual derivative. In particular, it is represented ... derivative is math Df a mathbf R n to mathbf R m quad mbox with quad Df a v J f a , v, math ... basis of R sup n sup . Since the derivative is a linear function, we have for all vectors h R sup n sup that the directional derivative of f along h is given by math Df a h sum i 1 n h i ... differentiable and yet fail to have continuous partial derivatives. Relation to the G teaux derivative A function f U V W is called G teaux derivative G teaux differentiable at x &isin U if f has a directional derivative along all directions at x . This means that there exists a function ... if its G teaux derivative is linear and bounded linear operator bounded at each point of U and the G teaux derivative is a continuous map nowrap U L V , W . cn date October 2011 For example, the real ... more details
real number line extended sense . See also Denjoy Young Saks theorem Derivative generalizations ... derivative first T.P. last Lukashenko year 2001 . Cite book first H.L. last Royden title Real analysis ... id 4714 title Dini derivative Category Generalizations of the derivative Category Real analysis ... more details
Context sate June 2011 date June 2011 In mathematics , the H derivative is a notion of derivative in the study of abstract Wiener space s and the Malliavin calculus . Definition Let math i H to E math be an abstract Wiener space, and suppose that math F E to mathbb R math is Fr chet derivative differentiable . Then the Fr chet derivative is a map math mathrm D F E to mathrm Lin E mathbb R math i.e., for math x in E math , math mathrm D F x math is an element of math E math , the dual space to math E math . Therefore, define the math H math derivative math mathrm D H F math at math x in X math by math mathrm D H F x mathrm D F x circ i H to R math , a continuous function continuous linear map on math H math . Define the math H math gradient math nabla H F E to H math by math langle nabla H F x , h rangle H left mathrm D H F right x h lim t to 0 frac F x t i h F x t math . That is, if math j E to H math denotes the adjoint of math i H to E math , we have math nabla H F x j left mathrm D F x right math . See also Malliavin derivative References unreferenced date June 2008 Category Generalizations of the derivative Category Measure theory Category Stochastic calculus probability stub ... more details
In number theory , the arithmetic derivative , or number derivative , is a function defined for integer s, based on prime factorization , by analogy with the product rule for the derivativederivative of a function that is used in mathematical analysis . Definition For natural numbers defined as follows math p 1 math for any prime math p math . math ab a b , ,ab math for any math a textrm , , b in mathbb N math product rule Leibniz rule . To coincide with the Leibniz rule math 1 math is defined to be math 0 math , as is math 0 math . Explicitly, assume that math x p 1 e 1 cdots p k e k textrm , math where math p 1, , dots, , p k math are distinct primes and math e 1, , dots, , e k math are positive integers. Then math x sum i 1 k e ip 1 e 1 cdots p i 1 e i 1 p i e i 1 p i 1 e i 1 cdots p k e k sum i 1 k e i frac x p i . math The arithmetic derivative also preserves the power rule for primes math p a ap a 1 textrm , math where math p math is prime and math a math is a positive integer. For example, math begin align 81 3 4 & 9 cdot 9 9 cdot 9 9 cdot 9 2 9 3 cdot 3 & 2 9 3 cdot 3 3 cdot 3 2 9 cdot 6 108 4 cdot 3 3. end align math The sequence of number derivatives for k 0, 1, 2, ... begins ... defines the derivative over the integers. Barbeau also further extended it to rational numbers. Victor ... and bounds E. J. Barbeau examined bounds of the arithmetic derivative. He found that the arithmetic derivative of natural numbers is bounded by math n leq frac n log k n k math where k is the least ... to find similar bounds for the arithmetic derivative extended to rational numbers by proving that between ... derivative , Canadian Mathematical Bulletin Vol. 4 1961 , 117 122. Victor Ufnarovski and Bo ... ArithmeticDerivative.html Arithmetic Derivative , Planet Math , accessed 04 15, 9 April 2008 UTC L. Westrick. http web.mit.edu lwest www intmain.pdf Investigations of the Number Derivative . Peterson ... of the arithmetic derivative . Category Number theory Category Generalizations of the derivative numtheory ... more details
In mathematics , the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from derivative calculus . Though they appear similar, the algebraic advantage of a formal derivative is that it does not rely on the notion of a limit mathematics limit , which is in general impossible to define for a ring mathematics ring . Many of the properties of the derivative are true of the formal derivative, but some, especially those that make numerical statements, are not. The primary use of formal differentiation in algebra is to test for multiple roots of a polynomial . Definition The definition of a formal derivative is as follows fix a ring R not necessarily commutative and let A R x be the ring of polynomials over R . Then the formal derivative is an operation on elements of A , where if math f x , ,a n x n cdots a 1 x a 0 math then its formal derivative is math f x , ,Df x n a n x n 1 cdots 2 a 2 x a 1 math just as for polynomials over the real numbers real or complex numbers complex numbers. Properties It can ... also be included as a linearity property. The formal derivative satisfies the Product rule Leibniz ... . Application to finding repeated factors As in calculus, the derivative detects multiple roots ... however, its formal derivative is zero since 3 0 in R and in any extension of R , so when we pass ... to analytic derivative When the ring R of scalars is commutative, there is an alternative and equivalent definition of the formal derivative, which resembles the one seen in differential calculus ... with the formal derivative of f as it was defined above. This formulation of the derivative ... continuous at math X math , it will recapture the classical definition of the derivative. If it is carried .... This way differentiation becomes a part of algebra of functions. See also Derivative ... derivative References Lang Algebra edition 3r http arxiv.org abs 0905.3611 Michael Livshits, You could ... more details
lowercase In mathematics , in the area of combinatorics , the q derivative , or Jackson derivative , is a q analog of the ordinary derivative , introduced by Frank Hilton Jackson . It is the inverse of Jackson integral Jackson s q integration Definition The q derivative of a function f x is defined as math left frac d dx right q f x frac f qx f x qx x . math It is also often written as math D qf x math . The q derivative is also known as the Jackson derivative . It is a linear operator math displaystyle D q f x g x D q f x D q g x . math It has product rule analogous to the ordinary derivative product rule which has two equivalent forms math displaystyle D q f x g x g x D q f x f qx D q g x g qx D q f x f x D q g x . math Similarly it satisfies a quotient rule math displaystyle D q f x g x frac g x D q f x f x D q g x g qx g x , quad g x g qx neq 0. math There is also a rule similar to the chain rule for ordinary derivatives. Let math g x c x k math . Then math displaystyle D q f g x D q k f g x D q g x . math Relationship to ordinary derivatives Q differentiation resembles ordinary differentiation, with curious differences. For example, the q derivative of the monomial is math left frac d dz right q z n frac 1 q n 1 q z n 1 n q z n 1 math where math n q math is the q bracket of n . Note that math lim q to 1 n q n math so the ordinary derivative is regained in this limit. The n th q derivative of a function may be given as math D n q f 0 frac f n 0 n frac q q n 1 q n frac f n 0 n n q math provided that the ordinary n th derivative of f exists at x 0. Here, math q q n math is the q Pochhammer symbol , and math n q math is the q factorial . If math f x math is analytic we can apply the Taylor formula to the definition of math D q f x math to get math displaystyle D q f x sum ... See also Derivative generalizations Jackson integral Q exponential Q difference polynomial s Quantum ... 9908140 A note on the q derivative operator , 1999 ArXiv math 9908140 Thomas Ernst, http www.math.uu.se ... more details
unreferenced date March 2010 In mathematics , a weak derivative is a generalization of the concept of the derivative of a function mathematics function strong derivative for functions not assumed differentiable , but only Integrable function integrable , i.e. to lie in the Lebesgue space math L 1 a,b math . See distribution mathematics distribution s for an even more general definition. Definition Let math u math be a function in the Lebesgue space math L 1 a,b math . We say that math v math in math L 1 a,b math is a weak derivative of math u math if, math int a b u t varphi t dt int a b v t varphi t dt math for all infinitely differentiable function s math varphi math with math varphi a varphi b 0 math . This definition is motivated by the integration technique of Integration by parts . Generalizing ... alpha math is a multi index , we say that math v math is the math alpha th math weak derivative of math ... in math U math . If math u math has a weak derivative, it is often written math D alpha u math since ... is not differentiable at t 0, has a weak derivative v known as the sign function given ... if t 0. end cases math This is not the only weak derivative for u any w that is equal to v almost everywhere is also a weak derivative for u . Usually, this is not a problem, since in the theory of Lp ... yet has a weak derivative. Since the Lebesgue measure of the rational numbers is zero, math int 1 mathbb Q t varphi t dt 0 math Thus math v t 0 math is the weak derivative of math 1 mathbb Q ... are equivalent if they are equal almost everywhere, then the weak derivative is unique. Also, if u is differentiable in the conventional sense then its weak derivative is identical in the sense given above to its conventional strong derivative. Thus the weak derivative is a generalization of the strong ... for the weak derivative. Extensions This concept gives rise to the definition of weak solution s in Sobolev ... functions Category Functional analysis Category Generalizations of the derivative ca Derivada ... more details
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