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Derivative of e^(x) evaluated at e

  1. Sep 17, 2007 #1
    1. The problem statement, all variables and given/known data
    Okay, so.. I'm confused about this problem, because I'm nearly certain that there's no "correct" answer from the options given.

    Consider the function f(x)=e^(x). Which of the following is equal to f'(e)? Note that there may be more than one


    2. Answer options
    a)
    Lim (e^(x+h))/h
    h->0

    b)
    Lim (e^(x+h)-e^(e))/h
    h->0

    c)
    Lim (e^(x+h)-e)/h
    h->0

    d)
    Lim (e^(x+h)-1)/h
    h->0

    e)
    Lim (e^(e)) * (e^(h)-1)/h
    h->0

    f)
    Lim e * (e^(h)-1)/h
    h->0

    3. The attempt at a solution
    Since f(x)=e^(x)
    f'(x)=e^(x) as well, and f'(e)=e^e

    From that, I'm pretty sure that options a-d are undefined, as you can't divide by 0.

    However, for option e: I got e^(e) - 1 as the answer (which isn't equal to f'(e))
    and for option f: I got e-1 as the answer.

    I'm confuzzled. Any help would be greatly appreciated!!!
     
    Last edited: Sep 17, 2007
  2. jcsd
  3. Sep 17, 2007 #2
    Well F is the right answer... Have you ever proved that

    [tex]\frac{d}{dx}e^x = e^x[/tex] just using the difference quotient?

    If you haven't with the limit laws we can re-write f as:

    [tex]\lim_{\substack{h\rightarrow 0}}e * \lim_{\substack{h\rightarrow 0}}\frac{e^h-1}{h} = \lim_{\substack{h\rightarrow 0}}e * 1 = e[/tex]
     
  4. Sep 17, 2007 #3
    What's the definition of a derivative and whats special about e?
     
  5. Sep 17, 2007 #4
    Sorry, that's what I meant instead of e-1.

    That part does equal e, but it has to equal e^(e) instead of just e, since f(x)=e^x, f '(x)= e^(x), so f '(e)=e^(e)

    Going by the limit laws, wouldnt the answer be option E?
    Since it's the same thing as F, but with e^e instead... sorry I can't make it pretty, I'm not used to this

    Thanks for helping...
     
    Last edited: Sep 17, 2007
  6. Sep 18, 2007 #5
    Oh shoot I'm sorry -- I wasn't thinking. The answer is letter e, but you still arrive at that answer by almost the same process as I just posted... factor out e^e and the everything else goes to 1. Does that make sense?
     
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