Explanation:
Slope
The
slope of a line in the plane is defined as the rise over the run, m = Δy/Δx.
In mathematics, the slope or gradient of a line describes its steepness, incline, or
grade. A higher slope value indicates a steeper incline.
The slope is (in the simplest of terms) the measurement of a line,
and is defined as the ratio of the "rise" divided by the
"run" between two points on a line, or in other words, the ratio of
the altitude change to the horizontal distance between any two points on the
line. Given two points (x_{1},y_{1}) and (x_{2},y_{2})
on a line, the slope m of the line is
Through differential
calculus, one can calculate the slope of the tangent
line to a curve at a point.
The concept of slope applies directly to grades or gradients in geography and civil
engineering. Through trigonometry,
the grade m of a road is related to its angle of
incline θ by
http://en.wikipedia.org/wiki/Slope
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.
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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above explanation is copied from Wikipedia, the free encyclopedia and is
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