The Derivative-----general overview
1) Once we have established the concept of
a limit, we can define the concept of continuity. For a function to be
continuous at x=a, we need to have a limit as x→a and that limit must
be
equal to the functional value at a, namely f(a).
2) Another major concept that depends on a
limit is the derivative. Once established, it will serve as the basis for one of the
two of the most important operations of all of calculus, namely
differentiation. Also, described as finding the instantaneous rate of change of a
function at a point.
Differentiation is the process of finding derivatives of functions.
Initially, this process is lengthy, but there are short cuts that reduce the time
involved considerable.
3) Prior to the concept of a limit, the
only type of rate of change that could be found was the average rate of change of a
function between two values of the independent variable. With
onset
of the derivative, we will have a way of narrowing our analysis to an isolated
value of the independent value. A very powerful
tool.
4) Before limits, the average rate of
change could be calculated by connecting two points on the function, drawing a
straight line (secant line), then finding the slope. Having knowledge of
limits
enables us to analyze the dynamic situation of letting one point approach the
other on the function & finding the limiting position of this
secant line. It would be the tangent line at one of the points. Hence,
the slope between two points, becomes the slope at a point, and the
average
rate of change between two points becomes the instantaneous rate of change at a
point.
5) Computing the derivative
entails setting up the limit that produces the above situation and
evaluating the result. The result will be known as the
derivative of the function. There are two main ways of setting up
this derivative limit, one gives a general derivative as a function of x
and the
other gives the numerical value of the derivative at a specified point. The
first form is mainly used in basic courses, while the second is very
handy in more advanced theoretical courses.
6) With the derivative, we are able to analyze & solve many problems dealing with dynamic situations in many different fields of study, including business & economics