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Definition of derivative

  1. Sep 4, 2006 #1
    Could som1 please tell me what the next steps would be to be able to remove the h in the denomenator. :confused:

    y = e^(7x+4)


    Definition: lim f(x+h) - f(x)
    h->0 h

    lim (e^(7(x+h) + 4) - (e^(7x+4))
    h->0 h

    lim (e^(7x + 7h + 4)) - (e^(7x +4))
    h->0 h
     
  2. jcsd
  3. Sep 4, 2006 #2

    HallsofIvy

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    How about using some of the properties of the exponential?
    [tex]e^{7x+7h+4}= e^{7h}e^{7x+ 4}[/tex]
    (yes, I could also have separated the "4" but it is the "h" that is important)
    so
    [tex]e^{7x+ 7h+ 4}- e^{7x+ 4}= e^{7x+4}(e^{7h}- 1)[/tex]
    You will still have to deal with
    [tex]\lim_{h\rightarrow 0}\frac{e^{7h}-1}{h}= 7\lim_{h\rightarrow 0}\frac{e^{7h}-1}{7h}[/tex]
    and, taking k= 7h,
    [tex] 7\lim_{k\rightarrow 0}\frac{e^{k}-1}{k}[/tex]

    but if you know how to deal with the derivative of ex you should be able to do that.
     
    Last edited: Sep 4, 2006
  4. Sep 4, 2006 #3
    i know that the final answer is 7 x .5 = 3.5 but i don't get how you got rid of e^(7x+4).

    e^(7x+7h+4) - e^(7x+4) = e^(7x+4)(e^(7h) - 1) and then somehow
    the e^(7x + 4) disappears and u get lim e^(7h-1)
    h->0 h
     
  5. Sep 4, 2006 #4
    Do you know how to derive the derivative of ex from the definition? If you can't, then you won't be able to solve this problem.
     
  6. Sep 4, 2006 #5

    HallsofIvy

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    If that is the answer, then what is the question?

    The derivative of e7x+4 is 7e7x+4! You don't "rid of" e7x+4, it's part of the answer. Since you assert that the answer is a number, 3.5, is it possible that the problem asks for the derivative at a given value of x?
     
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