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Definition of Derivative

  1. Aug 15, 2007 #1
    1. The problem statement, all variables and given/known data
    Find the derivative using the Definition of the Derivative:

    f(x) = 1 / x^2


    2. Relevant equations

    The Definition:

    f`(a) = lim h->0 [f(a+h) - f(a)] / h

    3. The attempt at a solution

    This is what I did:

    f`(a) = lim h->0 [tex](1/(x+h)^{2}) - 1/x^{2}) / (h)[/tex]

    f`(a) = lim h->0 [tex][((x^{2}) - 1 (x^{2} + 2xh + h^{2})) / x^{2}(x^{2} + 2xh + h^{2})] / h[/tex]

    f`(a) = lim h->0 [tex](2x + h) / (x^{4} + 2x^{3}h + x^{2}h^{2})[/tex]

    and finally

    f`(a) = lim h->0 [tex]2 / x^{3}[/tex]

    So I got that as the derivative, and if I did it correctly it should be right. Did I use the definition properly?
     
  2. jcsd
  3. Aug 15, 2007 #2
    lol nevermind im a retard i figured it out... sigh so much typing for nothing
     
  4. Aug 15, 2007 #3
    Should be

    [tex]f'(a) = \lim_{h \to 0} \frac{-(2x + h)}{(x^{4} + 2x^{3}h + x^{2}h^{2})}[/tex]

    Also when you get to the final answer [itex]\frac{-2}{x^3}[/itex] you already took the limit so the answer is just:

    [tex]f'(a) = \frac{-2}{x^3}[/tex]

    Because:
    [tex]f'(a) = \lim_{h \to 0} \frac{-(2x + 0)}{(x^{4} + 2x^{3}*0 + x^{2}*0^{2})}[/tex]
    [tex]f'(a) = \frac{-2x}{x^{4}}[/tex]
     
    Last edited: Aug 15, 2007
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