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Math Help - Solve using the quotient rule

  1. #1
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    Solve using the quotient rule

    Can someone please help me out with this.

    Differentiate square root of x divided by x^2+3


    The work i've done:


    1/2x^(-1/2) (x^2+3) - 2x (x^ 1/2) / (x^2+3)^2

    = 1/2x^(3/2) + 3/2x^(-1/2) - 2x^(3/2)

    = -3/2x^(3/2) + 3/2x^(-1/2) / (x^2+3)^2


    After that it's all blurry, not sure what to do....


    thanks for the help in advance.
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  2. #2
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    Quote Originally Posted by agent2421 View Post
    Can someone please help me out with this.

    Differentiate square root of x divided by x^2+3


    The work i've done:


    1/2x^(-1/2) (x^2+3) - 2x (x^ 1/2) / (x^2+3)^2

    = 1/2x^(3/2) + 3/2x^(-1/2) - 2x^(3/2)

    = -3/2x^(3/2) + 3/2x^(-1/2) / (x^2+3)^2


    After that it's all blurry, not sure what to do....


    thanks for the help in advance.
    \dfrac{d}{dx} \left(\dfrac{\sqrt{x}}{x^2+3}\right)

    \dfrac{(x^2+3) \cdot \frac{1}{2\sqrt{x}} - \sqrt{x} \cdot 2x}{(x^2+3)^2}

    \dfrac{(x^2+3) - 2x \cdot 2x}{2\sqrt{x}(x^2+3)^2}

    \dfrac{3(1 - x^2)}{2\sqrt{x}(x^2+3)^2}
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  3. #3
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    thanks... can you explain how you did the 3rd part though? When you brought 2 square root of x to the bottom... and multiplied the top by 2 x... i didn't get that part fully... also the last step where does 3 (1-x^2) come from?
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  4. #4
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    Skeeter multiplied the numerator and denominator by 2\sqrt{x} to clear that fraction in the numerator hence 2\sqrt{x} in the denominator goes straight in.

    The numerator is \left(2\sqrt{x} \times (x^2+3)\right) \cdot \dfrac{1}{2\sqrt{x}} - \left(2\sqrt{x} \cdot \sqrt{x} \cdot 2x \right)

     = (x^2+3) - 2x \cdot 2x = (x^2+3) - 4x^2 = 3-3x^2 = 3(1-x^2)
    Last edited by e^(i*pi); February 13th 2011 at 02:50 PM. Reason: making terms clearer
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  5. #5
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    2nd to 3rd step ...

    \dfrac{2\sqrt{x}}{2\sqrt{x}} \cdot \dfrac{(x^2+3) \cdot \frac{1}{2\sqrt{x}} - \sqrt{x} \cdot 2x}{(x^2+3)^2}

    \dfrac{(x^2+3) - 2x \cdot 2x}{2\sqrt{x}(x^2+3)^2}


    3rd to last step ... combine like terms and factor out the 3

    \dfrac{3(1 - x^2)}{2\sqrt{x}(x^2+3)^2}
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