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Thread: Derivative

  1. #1
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    Derivative

    Hi.
    How to find the derivative of 1/(x^2) using definition of the derivative;I know how to find it by using the power rule but not the definition.




    Thanks.
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  2. #2
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    The definition of the derivative is: If $\displaystyle f(x) = \frac{1}{x^2}$ then
    $\displaystyle f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}$
    So try doing that and come back and tell us how you did. We can help you if you still need help.
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  3. #3
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    I need a hint, 'cause I'm still stuck.
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  4. #4
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    Tell me what f(x+h) and f(x) are, then tell post what $\displaystyle \frac{f(x+h) - f(x)}{h}$. Once you have all that you should be able to simplify things and take the derivative.
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  5. #5
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    $\displaystyle \displaystyle f'(x) = \lim_{h \to 0} \frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h}$

    start by getting a common denominator and get those two fractions in the numerator together.
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  6. #6
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    I keep getting lost during simplifying process.(what's wrong with me)
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  7. #7
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    Quote Originally Posted by wowo View Post
    I keep getting lost during simplifying process. what's wrong with me
    can' say ... you haven't shown anything other than posting the problem.

    start by getting these two fractions together ...

    $\displaystyle \displaystyle \frac{1}{(x+h)^2} - \frac{1}{x^2}$

    ... note that the common denominator is $\displaystyle x^2(x+h)^2$.

    do it.
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  8. #8
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    $\displaystyle x^2-1(x^2+2xh+h^2)/(x^2+2xh+h^2)(x^2)/h$

    Am I right????
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  9. #9
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    Yes, continue simplification. Eventually an h will cancel.
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  10. #10
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    Quote Originally Posted by wowo View Post
    $\displaystyle x^2-1(x^2+2xh+h^2)/(x^2+2xh+h^2)(x^2)/h$

    Am I right????
    keep going ... clean it up
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  11. #11
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    Okay, I'm completely lost with simplifying
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  12. #12
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    Quote Originally Posted by wowo View Post
    Okay, I'm completely lost with simplifying
    $\displaystyle \displaystyle \frac{1}{h} \left[\frac{x^2 - (x^2 + 2xh + h^2)}{x^2(x+h)^2}\right]$

    distribute the negative in the numerator and combine like terms ... then factor out an $\displaystyle h$ from what's left in the numerator ...
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  13. #13
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    So, It's $\displaystyle 2x+h/(x^2(x+h)^2$
    $\displaystyle 2x/(x^2)(x^2)$
    $\displaystyle 2/x^3$
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  14. #14
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    Quote Originally Posted by wowo View Post
    So, It's $\displaystyle 2x+h/(x^2(x+h)^2$
    $\displaystyle 2x/(x^2)(x^2)$
    $\displaystyle 2/x^3$
    No. You have made a careless mistake in the numerator. Go back, check your work, find the mistake. Hint: You know what the final answer is meant to be, don't you ....?
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  15. #15
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    Is it -2x+h/(x^2(x+h)^2 ??
    Last edited by wowo; Oct 17th 2010 at 05:51 PM.
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