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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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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.


A substance that is made from another substance.
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.


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 ?
Marketing success through differentation of anything.
Chondrogenic differentation of clonal mouse embryonic cell line ATDC 5 in vitro: differentiation-dependent gene expression of parathyroid hormone (PTH)/PTH- related peptide receptor.
3 Orthoclase Plagioclase Hyperstene Magnetite Ilmenite Corindon Sample Nomative Differentation Calculated Calculated mineral index density, melt g/cc density 005E Quartz 91.
provide a detailed discussion of the trees' morphology, anatomy, phenology, ecology, distribution, taxonomic differentation, and uses, among other topics.
Adaptation and ecological differentation in wheat with special reference to geographical variation of growth habit and Vrn genotype.
Atlas of the Languages of Suriname (Leiden, 2002), 118-19; Norval Smith, "The History of the Surinamese Creoles II: Origin and Differentation," idem, 139-41, 146-47.
The differentation as to the particular publisher is printed on the front cover and on the dust-wrapper.
Third, the conceptualization of a diaspora as a unitary actor overlooks the internal differentation and heterogeneity found within ethnic communities.
The campaign, developed by roasters and retailers, promotes the classification and differentation of roasted coffee qualities into three categories--Traditional, Superior, and Gourmet.
Acquisitions that utilize excess capacity may establish a cost advantage for the firm while those that develop dynamic capabilities may establish a differentation advantage.

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