derivation

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 der= ivative to show that

f ‘ (x) = lim a^x. a^h – a^x/h

 &nbsp= ;         h–>0


b)Explain why it is also true that

f ‘ (x)= a^x lim  = a^h-1/h

          &nb= sp;       h–>0

 

c)Use computing technology and small values of h to estimate the valu= e of

L = lim a^h-1/h

      h–>0

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

<= br>D)Note that it would be ideal if the value of the limit L was 1, for the= n f would be a

particularly special function: its derivative would b= e simply a^x, which would mean

that its derivative is itself. By exp= erimenting 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

  &nbs= p;  h–>0
 
E) Compute ln(2) and ln(3). Wh= at does your work in (b) and (c) suggest is true about

d/dx [2^x] an= d d/dx [2^x].

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

e?

=

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