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Math Help - Trouble with Quotient Rule

  1. #1
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    Trouble with Quotient Rule



    I need to simplify it further, but before I do I checked it into Mathematica and it says the derivative is incorrect. What did I do wrong?
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  2. #2
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    Re: Trouble with Quotient Rule

    Your calculations are correct. If you simplify you should get
    1/(x^2 sqrt(1+x^2))
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  3. #3
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    Re: Trouble with Quotient Rule

    I find it to be a bit easier, though there is more factoring involved, if you divide by x first:
     \frac{d}{dx} \left ( - \frac{\sqrt{1 + x^2}}{x} \right ) = -\frac{d}{dx} \left ( \sqrt{ \frac{1}{x^2} + 1} \right )

    = - \frac{d}{dx} \left ( \left [ \frac{1}{x^2} + 1 \right ] ^{1/2} \right )

    Now for the derivative:
    = -\frac{1}{2} \cdot \left ( \frac{1}{x^2} + 1 \right ) ^{-1/2} \cdot \frac{-2}{x^3}

    = \frac{1}{x^3 \left ( \frac{1}{x^2} + 1 \right ) ^{1/2} }

    I find this result to be reasonably adequate, but your text decided differently and cleared the fraction in the denominator. So, using a little trick:
    = \frac{1}{x^2 \cdot x \left ( \frac{1}{x^2} + 1 \right ) ^{1/2} }

    = \frac{1}{x^2 \left ( x^2 \cdot \frac{1}{x^2} + x^2 \cdot 1 \right ) ^{1/2} }

    = \frac{1}{x^2 \left ( 1 + x^2 \right ) ^{1/2} }

    -Dan
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