derivative

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derivative:

see calculuscalculus,
branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value.
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Derivative

the fundamental concept of the differential calculus. It characterizes the rate of change of a function. The derivative is a function defined, for every x, as the limit of the ratio

if the limit exists. A function whose derivative exists is said to be differentiable.

Every differentiable function is continuous. The opposite assertion, however, is false. There even exist continuous functions that are nowhere differentiable. The derivative of a function of a real variable may be nondifferentiable or even discontinuous. In the complex domain, on the other hand, the existence of the first derivative implies the existence of derivatives of all orders. Derivatives of functions of more than one variable (partial derivatives), the rules for obtaining derivatives, and various applications of derivatives are discussed in the article DIFFERENTIAL CALCULUS.

The theory of functions of a real variable deals, in particular, with the functional properties of the derivative and with various generalizations of the concept of the derivative. For example, a derivative that exists everywhere is a function of class one in the Baire classification. A derivative, even if it is discontinuous, takes on all intermediate values between its maximum and minimum. The most important generalizations of the concept of derivative follow.

Dini derivatives. The superior limit of the ratio

as x1x, x1 > x, is called the right upper derivative of f Δd. The right lower λd and left upper Δs and lower λs derivatives are defined in an analogous manner. If Δd = λds = λs), f(x) has a right (left) derivative at the point x. The ordinary derivative exists if all four Dini derivatives are finite and equal. Dini derivatives were introduced by the Italian mathematician U. Dini in 1878. In 1915, N. N. Luzin proved that if all four Dini derivatives are finite on some set, then, apart from a null set, the function has an ordinary derivative everywhere on the set.

Approximate derivative. The approximate derivative was introduced by A. Ia. Khinchin in 1916. It is the limit of the ratio

as x1, approaches x on the points of a set for which x is a density point.

derivative

[də′riv·əd·iv]
(chemistry)
A substance that is made from another substance.
(mathematics)
The slope of a graph y = ƒ(x) at a given point c ; more precisely, it is the limit as h approaches zero of ƒ(c + h) - ƒ(c) divided by h. Also known as differential coefficient; rate of change.

derivative

1. Chem a compound that is formed from, or can be regarded as formed from, a structurally related compound
2. Maths
a. the change of a function, f(x), with respect to an infinitesimally small change in the independent variable, x; the limit of [f(a + Δx)--f(a)]/Δx, at x = a, as the increment, Δx, tends to 0. Symbols: df(x)/dx, fʹ(x), Df(x)
b. the rate of change of one quantity with respect to another
3. Psychoanal an activity that represents the expression of hidden impulses and desires by channelling them into socially acceptable forms
References in periodicals archive ?
All the same, courts should acknowledge that stockholders embark upon an alternative and potentially constructive kind of democratic engagement when they sue derivatively.
This is because in adopting a position concerning the metaphysical notion of change or constancy, a philosopher is ultimately declaring allegiance to a view of time (and, derivatively, a view of reality).
As an indication of the semantic framework's flexibility and substantive interest, I sketch some ethical theories that are naturally suggested by reflection on the semantic framework, and I show how these theories can themselves derivatively generate particular versions of such a framework.
A shortcoming of the work is that in developing his interpretation, Roark engages almost exclusively with a small handful of contemporary Aristotelian commentators, most notably Richard Sorabji (Time, Creation, and the Continuum, 1983) and Ursula Coope (Time for Aristotle, 2005), or, derivatively, with thinkers specifically treated in the works of these two.
A first problem is that it is not clear whether ownership should be understood derivatively in terms of epistemic authority, or as a feature of the overall causal process of disclosure of the environment, and, if the latter, it is not clear how Rowlands distinguishes between relevant and irrelevant causal contributions to disclosure without presupposing ownership.
Lewis says, "A form is what gives a compound material substance its determinate nature, and it may seem odd to find form picking up its character as a this derivatively from the fact that the compound is a this rather than the other way around" (p.
On Thursday, the Court ruled that "the undisputed facts establish that [Ohoven and Infinity] hold no copyright interest in the Screenplay -- or derivatively in the Film -- and that the Screenplay is not a joint work.
The major claim is that the English perfect is purely aspectual (a claim already made by Jespersen), and consequently it is not involved in the tenses system, except derivatively.
Finally, the author outlines why receptive passivity (and, thereby, the double direction of permeability) is crucial in order to identify the goal of psychoanalytic treatment and, derivatively, to understand how a certain kind of awareness may have a significant therapeutic effect.
Plaintiff and his wife, suing derivatively, commenced this action against, among others, Consolidated and Chroma, alleging violations of Labor Law [section] 240 (1) and [section] 241 (6).
Since several of these became available only when Loux was completing his book, it is a salient virtue thereof (and, derivatively, of its author) that it extensively engages these publications.
They reject the view that only believed propositions can be part of one's evidence, since they accept that experiences can be evidence and that, in fact, all ultimate evidence is experiential, so that beliefs are evidence only derivatively.

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