Section 3.3

You are responsible on a test for reproducing the proof of the formula (5) under Derivatives of Exponential Functions.  It is proved using logarithmic differentiation.

Although the following problems look the same, they are all done differently.
Find f '(x):

1. f(x) = 2^x
2. f(x) = x²
3. f(x) = 2³
4. f(x) = x^x

The first one uses formula (5) in the text.
Number two uses the power rule.
Number three is a constant function, so its derivative is zero.
Number four uses logarithmic differentiation.

A common error among students is to start using logarithmic differentiation all the time.  There are only three types of problems where you need logarithmic differentiation:

1)   Functions for which both the base and exponent contain the variable.

      e.g. f(x) = (sinx)^cosx

2)      Functions involving lots of powers, quotients, and products.

e.g.

3)      The proofs of formula (5) in this section and the power rule for real exponents from the previous section.

The author tells you in the homework directions when to use logarithmic differentiation.  I will not do this on a test.

You need to memorize the six derivatives below and need to be able to prove each fact if asked on a test. All 6 proofs are similar.

You do not need to memorize the range of inverse secant or inverse cosecant since it varies from text to text.  Since the range chosen by your author is the reason the last two derivatives have absolute values in the denominator, I will not mark off if you omit the absolute values.