Derivative
From KayakWiki
Derivative: In calculus, a derivative is a fundamental quantity that defines how a function is changing at a given point. In one dimension, the derivative of a continuous function f(x) is noted as f ′(x) and is defined as:
|
f ′(x) = |
lim |
( |
f(x+dx) - f(x) |
) |
|
dx→0 |
dx |
where the limit is taken as dx approaches zero.
What does this mean geometrically? Well think of a straight line. We can talk about the slope of that line as a measure of how steep the line is. Now consider a complex curve with lots of continuous wiggles in it (i.e. there are no breaks). We can talk about the slope of this curve at a given point as slope of a line tangent to the curve at that point. The definition of the derivative tells you how to find that tangent curve. You pick a pair of points around where you are interested in determining the derivative and draw a line through those points of the curve. Now move the two points closer and closer together (that's what it means to talk about the limit as dx approaches zero). When the separation between the points is vanishingly close to zero, you have a graphical representation of the derivative.
See also: Slope of the curve
