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# derivative

1. derived.
2. not original; secondary.
n.
3. something derived.
4. Also called derived form. Gram. a form that has undergone derivation from another, as atomic from atom.
5. Chem. a substance or compound obtained from, or regarded as derived from, another substance or compound.
6. Also called differential quotient; esp. Brit., differential coefficient. Math. the limit of the ratio of the increment of a function to the increment of a variable in it, as the latter tends to 0; the instantaneous change of one quantity with respect to another, as velocity, which is the instantaneous change of distance with respect to time. Cf. first derivative, second derivative.
7. a financial contract whose value derives from the value of underlying stocks, bonds, currencies, commodities, etc.
[1400-50; late ME derivatif < LL derivativus, equiv. to L derivat(us) (see DERIVATION) + -ivus -IVE]

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In mathematics, a fundamental concept of differential calculus representing the instantaneous rate of change of a function.

The first derivative of a function is a function whose values can be interpreted as slopes of tangent lines to the graph of the original function at a given point. The derivative of a derivative (known as the second derivative) describes the rate of change of the rate of change, and can be thought of physically as acceleration. The process of finding a derivative is called differentiation.

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in mathematics, the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations (differential equation). In general, scientists observe changing systems (dynamical systems (analysis)) to obtain the rate of change of some variable of interest, incorporate this information into some differential equation, and use integration techniques to obtain a function that can be used to predict the behaviour of the original system under diverse conditions.

Geometrically, the derivative of a function can be interpreted as the slope of the graph of the function or, more precisely, as the slope of the tangent line at a point. Its calculation, in fact, derives from the slope formula for a straight line, except that a limiting (limit) process must be used for curves. The slope is often expressed as the “rise” over the “run,” or, in Cartesian terms, the ratio of the change in y to the change in x. For the straight line shown in the figure—>, the formula for the slope is (y1 − y0)/(x1 − x0). Another way to express this formula is [f(x0 + h) − f(x0)]/h, if h is used for x1 − x0 and f(x) for y. This change in notation is useful for advancing from the idea of the slope of a line to the more general concept of the derivative of a function.

For a curve, this ratio depends on where the points are chosen, reflecting the fact that curves do not have a constant slope. To find the slope at a desired point, the choice of the second point needed to calculate the ratio represents a difficulty because, in general, the ratio will represent only an average slope between the points, rather than the actual slope at either point (see figure—>). To get around this difficulty, a limiting (limit) process is used whereby the second point is not fixed but specified by a variable, as h in the ratio for the straight line above. Finding the limit in this case is a process of finding a number that the ratio approaches as h approaches 0, so that the limiting ratio will represent the actual slope at the given point. Some manipulations must be done on the quotient [f(x0 + h) − f(x0)]/h so that it can be rewritten in a form in which the limit as h approaches 0 can be seen more directly. Consider, for example, the parabola given by x2. In finding the derivative of x2 when x is 2, the quotient is [(2 + h)2 − 22]/h. By expanding the numerator, the quotient becomes (4 + 4h + h2 − 4)/h = (4h + h2)/h. Both numerator and denominator still approach 0, but if h is not actually zero but only very close to it, then h can be divided out, giving 4 + h, which is easily seen to approach 4 as h approaches 0.

To sum up, the derivative of f(x) at x0, written as f′(x0), (df/dx)(x0), or Df(x0), is defined as

if this limit exists.

differentiation—i.e., calculating the derivative—seldom requires the use of the basic definition but can instead be accomplished through a knowledge of the three basic derivatives, the use of four rules of operation, and a knowledge of how to manipulate functions.

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Universalium. 2010.

Synonyms:

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• derivative — de·riv·a·tive 1 /də ri və tiv/ n: a contract or security that derives its value from that of an underlying asset (as another security) or from the value of a rate (as of interest or currency exchange) or index of asset value (as a stock index) ◇… …   Law dictionary

• Derivative — De*riv a*tive, a. [L. derivativus: cf. F. d[ e]rivatif.] Obtained by derivation; derived; not radical, original, or fundamental; originating, deduced, or formed from something else; secondary; as, a derivative conveyance; a derivative word. [1913 …   The Collaborative International Dictionary of English

• Derivative — De*riv a*tive, n. 1. That which is derived; anything obtained or deduced from another. [1913 Webster] 2. (Gram.) A word formed from another word, by a prefix or suffix, an internal modification, or some other change; a word which takes its origin …   The Collaborative International Dictionary of English

• derivative — [adj] borrowed, transmitted from source acquired, ancestral, caused, cognate, coming from, connate, copied, evolved, hereditary, imitative, inferential, inferred, not original, obtained, plagiaristic, plagiarized, procured, rehashed, secondary,… …   New thesaurus

• derivative — ► ADJECTIVE 1) chiefly derogatory imitative of the work of another artist, writer, etc. 2) (of a financial product) having a value deriving from an underlying variable asset. ► NOUN 1) something which is derived from another source. 2) a… …   English terms dictionary

• derivative — [də riv′ə tiv] adj. [ME derivatif < LL derivativus < L derivatus, pp. of derivare: see DERIVE] 1. derived 2. using or taken from other sources; not original 3. of derivation n. 1. something derived 2 …   English World dictionary

• derivative — early 15c. (adj.); mid 15c. (n.), from M.Fr. dérivatif (15c.), from L.L. derivativus, from pp. stem of L. derivare (see DERIVE (Cf. derive)). Mathematical sense is from 1670s …   Etymology dictionary

• Derivative — This article is an overview of the term as used in calculus. For a less technical overview of the subject, see Differential calculus. For other uses, see Derivative (disambiguation) …   Wikipedia

• derivative — Coming from another; taken from something preceding; secondary. That which has not its origin in itself, but owes its existence to something foregoing. Anything obtained or deduced from another @ derivative action A suit by a shareholder to… …   Black's law dictionary

• derivative — [[t]dɪrɪ̱vətɪv[/t]] derivatives 1) N COUNT A derivative is something which has been developed or obtained from something else. ...a poppy seed derivative similar to heroin... This isn t an entirely new car, but a new derivative of the Citroen XM …   English dictionary