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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