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)
= x^{1 / 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
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