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Application of Derivative:Find Rate of Change

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Explanation:

 

Derivative

 

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity (conversely, integrating a car's velocity over time yields the distance traveled).

The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely related notion is the differential of a function.

The process of finding a derivative is called differentiation.

 

Differentiation and the derivative

 

Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means that y is a function of x. This functional relationship is often denoted y = ƒ(x),where ƒ denotes the function. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.

The simplest case is when y is a linear function of x, meaning that the graph of y against x is a straight line. In this case, y = ƒ(x) = m x + b, for real numbers m and b, and the slope m is given by

where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in."

 

Computing the derivative

The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones.

 

Derivatives of elementary functions

Most derivative computations eventually require taking the derivative of some common functions. The following incomplete list gives some of the most frequently used functions of a single real variable and their derivatives.

       Derivatives of powers:if

where r is any real number, then

wherever this function is defined. For example, if f(x) = x1 / 4, then

and the derivative function is defined only for positive x, not for x = 0. When r = 0, this rule implies that f′(x) is zero for x ≠ 0, which is almost the constant rule.

  Constant rule:if ƒ(x) is constant, then

(Our solved example in mathguru.com uses this concept)

 

http://en.wikipedia.org/wiki/Derivative

 

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.