Using these last few propositions we can prove many of the kinds of
identities that one comes across as exercises in differentiation in calculus.
Many of the results that we remember from calculus can be derived from
the results we already have. The Chain Rule, the Product Rule and the
Quotient Rule are the real mainstays of differentiation in calculus
class, and we have these in their final forms. The problem we are still
faced with, as mentioned at the end of the last class, is that we still
don't know very much about some of the standard functions of analysis:
the exponential functions and logarithms. Although as a result of the
digression on exponential functions (and Homework 6) we now know quite a
lot about the exponential and logarithmic functions, there are still
huge gaps in our knowledge.
We can't differentiate a^x or even x^b (unless
b is an integer).
Important rules from calculus are missing, such as f'(x) positive,
implies f(x) is increasing. That means we can't make use of
differentiation to find local maxima and minima yet.
We'll come back to these in the end of
The final section in this chapter deals with variations on the theme of
limits of functions: left-hand and right-hand limits, and limits as
x approaches infinity.