Derivative homework help

Let f(x) = a^x. The goal of this problem is to explore how the value of a affects the derivative
of f(x), without assuming we know the rule for d/dx [a^x] that we have stated and used in earlier
work in this section.

a) Use the limit definition of the derivative to show that
f ' (x) = lim a^x. a^h - a^x/h
           h-->0

b)Explain why it is also true that
f ' (x)= a^x
lim   a^h-1/h
                  h-->0

c)Use computing technology and small values of h to estimate the value of
L = lim a^h-1/h
h-->0

when a = 2. Do likewise when a = 3.

D)Note that it would be ideal if the value of the limit L was 1, for then f would be a
particularly special function: its derivative would be simply a^x, which would mean
that its derivative is itself. By experimenting with different values of a between 2 and
3, try to find a value for a for which:

L = lim a^h-1/h=1
h-->0

E) Compute ln(2) and ln(3). What does your work in (b) and (c) suggest is true about
d/dx [2^x] and d/dx [2^x].

F) How do your investigations in (d) lead to a particularly important fact about the number
e?