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Math Help - Multivariable calculus - implicit differentiation

  1. #1
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    Multivariable calculus - implicit differentiation

    Guys l just want to know if l am on the right path in this question. The question and attempt to the solution is on the pdf file attached.
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  2. #2
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    Just use the chain rule.

    You should know that \frac{du}{dx} = \frac{du}{dt}\,\frac{dt}{dx}.


    Since

    x^3t + x^2t - 4 = 0

    \frac{d}{dx}(x^3t + x^2t - 4) = \frac{d}{dx}(0)

    \frac{d}{dx}(x^3t) + \frac{d}{dx}(x^2t) + \frac{d}{dx}(-4) = 0

    x^3\,\frac{dt}{dx} + 3x^2t + x^2\,\frac{dt}{dx} + 2xt + 0 = 0

    (x^3 + x^2)\,\frac{dt}{dx} + 3x^2t + 2xt = 0

    x^2(x + 1)\,\frac{dt}{dx} + xt(3x + 2) = 0

    x^2(x + 1)\,\frac{dt}{dx} = -xt(3x + 2)

    \frac{dt}{dx} = -\frac{xt(3x + 2)}{x^2(x + 1)}

    \frac{dt}{dx} = -\frac{t(3x + 2)}{x(x + 1)}.


    You also know that

    u = e^{5 + t} + \cos{t}

    so \frac{du}{dt} = e^{5 + t} - \sin{t}.



    Therefore

    \frac{du}{dx} = \frac{du}{dt}\,\frac{dt}{dx}

     = -\frac{t\left(e^{5 + t} - \sin{t}\right)(3x + 2)}{x(x + 1)}.
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  3. #3
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    Quote Originally Posted by Prove It View Post
    Just use the chain rule.

    You should know that \frac{du}{dx} = \frac{du}{dt}\,\frac{dt}{dx}.


    Since

    x^3t + x^2t - 4 = 0

    \frac{d}{dx}(x^3t + x^2t - 4) = \frac{d}{dx}(0)

    \frac{d}{dx}(x^3t) + \frac{d}{dx}(x^2t) + \frac{d}{dx}(-4) = 0

    x^3\,\frac{dt}{dx} + 3x^2t + x^2\,\frac{dt}{dx} + 2xt + 0 = 0

    (x^3 + x^2)\,\frac{dt}{dx} + 3x^2t + 2xt = 0

    x^2(x + 1)\,\frac{dt}{dx} + xt(3x + 2) = 0

    x^2(x + 1)\,\frac{dt}{dx} = -xt(3x + 2)

    \frac{dt}{dx} = -\frac{xt(3x + 2)}{x^2(x + 1)}

    \frac{dt}{dx} = -\frac{t(3x + 2)}{x(x + 1)}.


    You also know that

    u = e^{5 + t} + \cos{t}

    so \frac{du}{dt} = e^{5 + t} - \sin{t}.



    Therefore

    \frac{du}{dx} = \frac{du}{dt}\,\frac{dt}{dx}

     = -\frac{t\left(e^{5 + t} - \sin{t}\right)(3x + 2)}{x(x + 1)}.

    Was the method l used correct ?
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  4. #4
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    Yes. Once you complete those determinants, you should get the same answer.
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