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Math Help - Simplyfy the following complex rational expressions 2

  1. #1
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    Simplyfy the following complex rational expressions 2

    Q)  \frac{1/(x+h)^2-1/x^2}{h}

    Please write the answer in steps thank you
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  2. #2
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    Quote Originally Posted by mj.alawami View Post
    Q)  \frac{1/(x+h)^2-1/x^2}{h}

    Please write the answer in steps thank you
    \frac{1}{h}\left[\frac{1}{(x+h)^2} - \frac{1}{x^2}\right]

    \frac{1}{h}\left[\frac{x^2}{x^2(x+h)^2} - \frac{(x+h)^2}{x^2(x+h)^2}\right]

    \frac{1}{h}\left[\frac{x^2 - (x+h)^2}{x^2(x+h)^2}\right]

    \frac{1}{h}\left[\frac{x^2 - (x^2+2xh+h^2)}{x^2(x+h)^2}\right]<br />

    \frac{1}{h}\left[\frac{-2xh-h^2}{x^2(x+h)^2}\right]

    \frac{1}{h}\left[\frac{-h(2x+h)}{x^2(x+h)^2}\right]

    -\frac{2x+h}{x^2(x+h)^2}
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  3. #3
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    =  \frac{ \frac{x^2 - (x+h)^2 } {(x+h)^2x^2}}{h}

     <br />
(x+h)^2 = x^2 +h^2 +2hx<br />

    = \frac{ \frac{x^2 - x^2-h^2-2hx} {(x+h)^2x^2}} {h}

     <br />
= \frac{ \frac{-h^2-2hx} {(x+h)^2x^2} } {h}<br />

     <br /> <br />
= \frac{-h(h+2x)} {h\times (x+h)^2x^2}<br />

    = \frac{-(h+2x)} { (x+h)^2x^2}
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  4. #4
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    Quote Originally Posted by mj.alawami View Post
    Q)  \frac{1/(x+h)^2-1/x^2}{h}

    Please write the answer in steps thank you
     \dfrac{\frac1{(x+h)^2}-\frac1{x^2}}{h}

    The denominator is x^2(x+h)^2:

     \dfrac{\frac1{(x+h)^2}-\frac1{x^2}}{h} = \dfrac{\frac{x^2}{x^2 (x+h)^2}-\frac{(x+h)^2}{x^2(x+h)^2}}{h} = \dfrac{x^2-(x+h)^2}{h\cdot x^2(x+h)^2}

    Expand the bracket in the numerator and collect like terms:

    \dfrac{x^2-(x+h)^2}{h\cdot x^2(x+h)^2} = \dfrac{-2hx-h^2}{h\cdot x^2(x+h)^2}

    Factor h out in the numerator and cancel:

     \dfrac{\frac1{(x+h)^2}-\frac1{x^2}}{h} = \dfrac{-2x-h}{x^2(x+h)^2}

    It seems to me as if you were asked to calculate the drivation of f(x)=\dfrac1x^2

    If so:

    f'(x)=\lim_{h\to 0}\left( \dfrac{-2x-h}{x^2(x+h)^2}\right) = \dfrac{-2x}{x^4} = \dfrac{-2}{x^3}
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