Once again, we review this wonderful wedding of algebra, limits, and geometry
which produces the derivative. Let's review this process by finding the derivative
of: .
Given function for which we are to find the derivative.
We are to find:
Here we are using the definition for a derivative that we have developed
in the previous lessons.
We follow our definition as it is applied to our particular given function.
Here we expand the powers.
Multiply and gather like terms.
Factor an h out of the numerator and cancel with the h
in the denominator.
Finally, we can take the limit, and we have our derivative.
The derivative as a rate of change. Let's solve this example problem: 2.7,
#30.
Example problem: 2.7, #30.
f '(8) signifies the sales rate in lbs./$ in cost when the cost of
gourmet coffee is $8/lb.?
Remember, derivatives are instananeous rates of change. So f '(8)
is really indicating the rate of change of the sales at the instant when
the number of pounds is 8.
We cannot tell if f '(8) is positive or negative.
We have not been given enough information about f(p). My guess
is that it is negative. Reasoning: Not too many people buy 8 pounds of
coffee at one time, therefore the cost is probably decreasing with more
than 5 or so lbs.
. 
Definition of a Derivative Teaching Video
