Let f(x) = a^x. The goal of this problem is to explore how the value of a affects the derivativeof f(x), without assuming we know the rule for d/dx [a^x] that we have stated and used in earlierwork in this section. a) Use the limit definition of the derivative to show thatf ? (x) = lim a^x. a^h a^x/h h>0 b)Explain why it is also true thatf ? (x)= a^x lim a^h-1/h h>0 c)Use computing technology and small values of h to estimate the value ofL = lim a^h-1/hh>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 aparticularly special function: its derivative would be simply a^x, which would meanthat its derivative is itself. By experimenting with different values of a between 2 and3, try to find a value for a for which: L = lim a^h-1/h=1h>0 E) Compute ln(2) and ln(3). What does your work in (b) and (c) suggest is true aboutd/dx [2^x] and d/dx [2^x]. F) How do your investigations in (d) lead to a particularly important fact about the numbere?
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