Derivative (redirected from Instantaneous rate of change)

derivative: see calculus.

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derivative

In mathematics, a fundamental concept of differential calculus representing the instantaneous rate of change of a function. The first derivative of a function is a function whose values can be interpreted as slopes of tangent lines to the graph of the original function at a given point. The derivative of a derivative (known as the second derivative) describes the rate of change of the rate of change, and can be thought of physically as acceleration. The process of finding a derivative is called differentiation.

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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

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derivative [də′riv·əd·iv]

(chemistry)

A substance that is made from another substance.

(mathematics)

The slope of a graphy = ƒ(x) at a given pointc; more precisely, it is the limit ashapproaches zero of ƒ(c + h) - ƒ(c) divided byh. Also known as differential coefficient; rate of change.

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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 x₁ →x, x₁ > x, is called the right upper derivative of f Δ_d. The right lower λ_d and left upper Δ_s and lower λ_s derivatives are defined in an analogous manner. If Δ_d = λ_d (Δ_s = λ_s), 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.