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Math Help - implicit differentiation

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    implicit differentiation

    y'' of xcosy = y
    find second derivative?
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  2. #2
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    Quote Originally Posted by RiemannManifold View Post
    y'' of xcosy = y
    find second derivative?
    x\cos{y} = y

    \frac{d}{dx}(x\cos{y}) = \frac{d}{dx}(y)

    x\frac{d}{dx}(\cos{y}) + \cos{y}\frac{d}{dx}(x) = \frac{dy}{dx}

    x\frac{d}{dy}(\cos{y})\frac{dy}{dx} + \cos{y} = \frac{dy}{dx}

    -x\sin{y}\frac{dy}{dx} + \cos{y} = \frac{dy}{dx}

    \cos{y} = \frac{dy}{dx} + x\sin{y}\frac{dy}{dx}

    \cos{y} = \frac{dy}{dx}(1 + x\sin{y})

    \frac{dy}{dx} = \frac{\cos{y}}{1 + x\sin{y}}.



    \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dx}\left(\frac{\cos{y}}{1 + x\sin{y}}\right)

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


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