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.
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."
Definition via
difference quotients
Rate of change as a limiting
value
Figure
1. The tangent line at (x, ƒ(x))
Figure
2. The secant to curve y= ƒ(x) determined by points (x, ƒ(x)) and (x+h, ƒ(x+h))
Figure
3. The tangent line as limit of secants
Let ƒ be a real valued function. In
classical geometry, the tangent line to the graph of the function ƒ at a real number a was the unique line through the point
(a, ƒ (a))
that did not meet the graph of ƒ transversally, meaning that the line did not pass straight through the graph.
The derivative of y with respect to x at a is, geometrically, the slope of the
tangent line to the graph of ƒ at a.
The slope of the tangent line is very close to the slope of the line through (a, ƒ(a)) and a nearby point
on the graph, for example (a + h, ƒ(a + h)).
These lines are called secant lines. A value of h close to zero gives a good
approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations. The slope m of the secant line is the difference
between the y values of these points divided by the
difference between the x values, that is,
This expression is Newton's difference quotient. The
derivative is the value of the difference quotient as the secant lines approach
the tangent line. Formally, the derivative of the function ƒ at a is the limit
of the difference quotient as h approaches zero, if
this limit exists. If the limit exists, then ƒ is differentiable at a. Here ƒ′ (a) is
one of several common notations for the derivative.
(Our solved
example in mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Derivative
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