# 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 ?
En efecto, si existen pacientes que tienen dificultades en los mecanismos flexivos y no en los derivativos, esta hipotesis de distincion entre ambos procesos morfologicos cobraria fuerza, mas aun si encontramos disociaciones dobles, como ocurre con Laudanna, Badecker y Caramazza (1992).
O sea que el aspecto atributivo reune una porcion de la base nominal que compone al diminutivo, a la que se adosa el derivativo (5):
Em um periodo de tempo pequeno, a variacao do preco do ativo e do derivativo sao perfeitamente correlacionados.
Ventana parenquimatosa con dilatacion del ventriculo lateral izquierdo a predominio del asta frontal extendida hasta el area de solucion de continuidad fronto-parietal izquierdo, ventriculo lateral derecho normo-configurado, se observa imagen tubular en el asta frontal del ventriculo lateral izquierdo en relacion con sistema derivativo cistoperitoneal, adecuada diferenciacion cortico subcortical, surcos y cisuras visibles, cisternas basales amplias y permeables.
Como seres expuestos, los peligros que tememos y, los miedos derivativos asociados a ellos, pueden ser, segun Bauman (2007):
El estudio de las estrategias de control en cascada, basado en acciones de control Proporcional, Integral y Derivativo, PID, es un tema habitual en cursos de control en Ingenieria y abordado por diferentes autores, quienes presentan diversas metodologias de implementacion para esta estrategia en plantas de intercambio termico.
Luego de integrar todos los bloques necesarios que intervienen en el sistema y sus respectivos parametros, con la opcion de sintonizacion del control PID se calculo los valores proporcional, integral y derivativos del sistema (tabla 5), en donde se puede observar que en la simulacion no se tiene un valor derivativo del controlador, ya que el crecimiento de la temperatura es lento en funcion del tiempo, por lo cual se deduce que esta parte no es necesaria ya que teoricamente no existe un error de estado estacionario, es decir, no es necesaria la estabilidad en lazo cerrado.
Ancora-se esse fundamento da derivacao com base no tema em uma relacao implicacional: em se considerando que o significado de um derivativo e composicional, entao e mais provavel que sua base seja um tema em vez de uma raiz.
Sin embargo, estudios recientes reportan hasta un 32% de disfunciones del sistema derivativo, siendo la principal causa la obstruccion del cateter ventricular por el plexo coroides (3,4,5).
Cuando hace una breve comparacion de 1812 con la primera Constitucion revolucionaria francesa de 1791, llega a la conclusion de que "no todo es derivativo en Cadiz" (p.

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