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Math Help - implicit differentiation problem

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    implicit differentiation problem

    If and , find by implicit differentiation.
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    Quote Originally Posted by tbenne3 View Post
    If and , find by implicit differentiation.
    \frac{1}{x} + \frac{1}{y} = 2

    x^{-1} + y^{-1} = 2

    \frac{d}{dx}(x^{-1} + y^{-1}) = \frac{d}{dx}(2)

    -x^{-2} + \frac{d}{dy}(y^{-1})\,\frac{dy}{dx} = 0

    -\frac{1}{x^2} - \frac{1}{y^2}\,\frac{dy}{dx} = 0

    -\frac{1}{y^2}\,\frac{dy}{dx} = \frac{1}{x^2}

    \frac{dy}{dx} = -\frac{y^2}{x^2}.


    You also know that when x = 4, y = \frac{4}{7}.


    So \frac{dy}{dx}|_{x = 4} = -\frac{\left(\frac{4}{7}\right)^2}{4^2}

     = -\frac{\frac{16}{49}}{16}

     = -\frac{1}{49}.
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    thanks but one question.. in step 3 how did you get the -x^2?
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    Quote Originally Posted by tbenne3 View Post
    thanks but one question.. in step 3 how did you get the -x^2?
    What's the derivative of x^{-1}?
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