Information AboutDerivative |
| CATEGORIES ABOUT DERIVATIVE | |
| mathematical analysis | |
| differential calculus | |
| functions and mappings | |
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In Mathematics , the derivative is defined as the ''instantaneous rate of change'' of a function. The derivative is one of the two central concepts of Calculus . (The other is the Integral ; the two are related via the Fundamental Theorem Of Calculus .) The simplest type of derivative is the derivative of a real-valued Function of a single Real variable. It has several interpretations:
This derivative is the kind usually encountered in a first course on calculus, and historically was the first to be discovered. However, there are also many Generalizations Of The Derivative . The remainder of this article discusses only the simplest case (real-valued functions of real numbers). DIFFERENTIATION AND DIFFERENTIABILITY In physical terms, differentiation expresses the rate at which a quantity, ''y'', changes with respect to the change in another quantity, ''x'', on which it has a Functional Relationship . Using the symbol Δ to refer to change in a quantity, this rate is defined as a Limit of difference quotients : as ''x'' approaches 0. In Leibniz's Notation For Derivatives , the derivative of ''y'' with respect to ''x'' is written : suggesting the ratio of two Infinitesimal quantities. The above expression is pronounced in various ways such as "''dy by dx''" or "''dy over dx''". The form "''dy dx''" is also used conversationally, although it may be confused with the notation for element of area. Modern mathematicians do not bother with "dependent quantities", but simply state that differentiation is a mathematical Operation on functions. One precise way to define the derivative is as a limit: : A function is differentiable at a point ''x'' if the above limit exists (as a finite real number) at that point; a function is differentiable on an Interval if it is differentiable at every point within the interval. As an alternative, the development of Nonstandard Analysis in the 20th century showed that Leibniz's original idea of the derivative as a ratio of infinitesimals can be made as rigorous as the formulation in terms of limits. |
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