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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 ?
Languages provide considerable evidence that the distribution of individual syntactic patterns may depend on factors other than the meaning inherently or derivatively associated with these patterns.
bringing a claim derivatively must be a shareholder at the time of the
I propose that states enact statutes granting current university students standing to sue their boards of trustees derivatively.
Jews and Christians share together, in starkly unequal and incommensurate ways, a mutual history of misunderstanding, distortion, and stereotype that has had disastrous effects upon both communities and, derivatively, upon the larger culture that Judaism and Christianity together inhabit.
Once the Fed pre-determines the value of "P", it is left to the banks to derivatively determine the value of "[beta]".
That general right derivatively protects personally important activities, and owning guns is, for many people, very important.
And, it certainly would not do Erin well, and derivatively you well, if a lot of negative facts about .
Money that once went for higher education in California, and derivatively for library expenses, is now going for the construction and upkeep of prisons.
Furthermore, if our emphasis as member churches in a council is first on Christ -- on serving our common Lord -- and only derivatively on each other, then to break whatever minimal fellowship we have achieved is indeed a great scandal.
Nevertheless, today so soon after his death, we must recognize and acknowledge Laurens van der Post, not only as a vivid and colourful travel writer, but more especially as - derivatively and cumulatively - a global and universal philosopher.
Moreover, he claimed rent stabilized status for his wife derivatively.
However, another verb sa'ada ("to help or assist"), is derivatively related to sa 'adah, the implication being that happiness is not totally an individual effort, but also depends on helping, or being helped by other members of the community.

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