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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 ?
t] and twice continuous differentiability in all of its arguments.
Assuming sufficient differentiability of u, this condition is equivalent to u'''' [less than or equal to] 0.
Note that, since the canonical metric, on account of its own definition, is conceived outside the matter, we have not to bother ourselves about questions of differentiability on the subspace R x {(0, 0, 0)} of R x [R.
relating to continuity, differentiability and Riemann integrability and also p roved the following theorems:
Use the strips of counties next to each legend to evaluate color differentiability.
The differentiability assumptions on which it rested did not allow, for instance, for consumers who do not consume some commodities.
These relate to the nonemptiness, cardinality and differentiability of the solutions with respect to the various parameters of interest.
Stahl develops the basic tools of advanced calculus, which introduce the various aspects of the completeness of the real number system as well as sequential continuity and differentiability and lead to the Intermediate and Mean Value Theorems.
From the monotonicity and the Frechet differentiability of F, it follows that (1.
First we prove the differentiability of z(u) at [alpha].
Among them are a simple numerical approach to the Riemann hypothesis, aunifying construction for measure-valued continuous and discrete branching processes, examples of quantitative universal approximation, harmonic mappings with quadilateral image, meromorphic approximation on noncompact Riemann surfaces, a family of outer functions, the universality of series in Banach space, recent progress on fine differentiability and fine harmonicity, reversibility questions in groups arising from analysis, and the generalized binomial theorem.
In this section, we state two theorems regarding continuous dependence and differentiability of solutions of (1.