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Proof of the Power Rule - Stuck at the End

  1. Jan 20, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove that the following

    [itex]f'(x)=nx^{n-1}[/itex] if [itex]f(x)=x^{n}[/itex]


    2. Relevant equations
    Binomial theorem, definition of the derivative


    3. The attempt at a solution

    [tex]f'(x)=lim_{h\rightarrow0}\frac{f((x+h)^{n})-f(x)}{h}[/tex]

    We need to expand the (x+h)^2 term now

    [tex]\sum^{n}_{k=0}{n\choose k} x^{n-k}h^{k}={n\choose 0}x^n+{n\choose 1}x^{n-1}h+...+{n\choose k}x^{n-k}h^{k}[/tex]

    So, we sub this for the f((x+h)^n) term:

    [tex]lim_{h \rightarrow 0}\frac{{n\choose 0}x^n+{n\choose 1}x^{n-1}h+...+{n\choose k}x^{n-k}h^{k}-x}{h}[/tex]

    [tex]{n\choose k}=\frac{n!}{(n-k)!k!}[/tex]

    This now simplifies to---

    [tex]lim_{h\rightarrow0}\frac{x^{n}}{h}+nx^{n-1}-lim_{h\rightarrow0}\frac{x}{h}[/tex]

    The first and third terms create only a one sided limit, and they both go to infinity, I'm not sure where I went wrong....

    I could just look up the proof, but I'm trying to do it by myself, so I'm only looking for a hint.

    I feel like I'm really close, because I have the answer there, I just don't know why those other two terms are messing it up.

    There are only two possibilities, either I've gone in the completely wrong direction, or I made a silly mistake somewhere.
     
    Last edited: Jan 20, 2013
  2. jcsd
  3. Jan 20, 2013 #2

    SteamKing

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    Your definition of f'(x) is a little strange.

    Since f(x) = x^n, then

    f'(x) = lim (h -> 0) [(x+h)^n - x^n) / h]

    Try expanding now using the Binomial Theorem.
     
  4. Jan 20, 2013 #3
    :bugeye:

    Thanks, I didn't catch that.... how silly.

    So, that means those two terms cancel out, and we're left with the correct one.
     
  5. Jan 20, 2013 #4

    LCKurtz

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    What do you mean by "we're left with the correct one"? There are two terms that do cancel but there are a lot of others and you have to deal with the h's. Have you actually worked it out?
     
  6. Jan 20, 2013 #5
    Yes, I skipped steps out of laziness (meaning, I did them on paper, but I didn't feel like taking the time to write them out here).

    The h on the bottom cancels out and all the hs on the top go to 0.
     
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