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Math Help - Hard derivative

  1. #1
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    Hard derivative

    Differentiate the fnction
    f(x) = \frac{(x^{-1} + x^{2})^{-1}}{(x^{3} - 2x^{-2})^{-2}}
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  2. #2
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    Differentiate the fnction
    f(x) = \frac{(x^{-1} + x^{2})^{-1}}{(x^{3} - 2x^{-2})^{-2}}
    Someone wants to give you an ulcer, don't they?

    Simplify first:

    f(x) = \frac{(x^{-1} + x^{2})^{-1}}{(x^3 - 2x^{-2})^{-2}}

    f(x) = \frac{(x^3 - 2x^{-2})^2}{x^{-1} + x^2}

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

    I'm going to expand the numerator so my next step will be a bit clearer:
    f(x) = \frac{x^6 - \frac{4x^3}{x^2} + \frac{4}{x^4}}{\frac{1}{x} + x^2}

    f(x) = \frac{x^6 - 4x + \frac{4}{x^4}}{\frac{1}{x} + x^2}

    Now mulitply the numerator and denominator by x^4. We would normally need to be wary of doing this, but the domain of the original function excludes x = 0 anyway in this case.
    f(x) = \frac{x^{10} - 4x^5 + 4}{x^3 + x^6}

    You should be able to take the derivative now.

    -Dan
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