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Math Help - derivative

  1. #1
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    derivative

    1) z=\frac{1}{\sqrt{x^2+4y^2}}

    a) \frac{\partial^2 z}{\partial x^2}

    b) \frac{\partial^2 z}{\partial x \partial y}

    The answer is???
    a) -\frac{\sqrt{x^2+4y^2}+2x^2}{x^2+4y^2}

    b) -\frac{4xy}{\sqrt{x^2+4y^2}}


    2)
    z=ln(x^2+y^2)
    a) \frac{\partial^3 z}{\partial x \partial y^2}

    The answer is????


    a) \frac{4x^2+4y^2-8x^3-8xy^2}{(x^2+y^2)^3}


    3) z=\sqrt{2xy+y}
    a) \frac{ \partial^2 z}{ \partial x \partial y}

    b) \frac{ \partial^3 y}{ \partial x^3}

    How I do the 3?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Apprentice123 View Post
    1) z=\frac{1}{\sqrt{x^2+4y^2}}

    a) \frac{\partial^2 z}{\partial x^2}

    b) \frac{\partial^2 z}{\partial x \partial y}

    The answer is???
    a) -\frac{\sqrt{x^2+4y^2}+2x^2}{x^2+4y^2}
    nope

    begin by writing the original as z = (x^2 + 4y^2)^{-1/2}

    b) -\frac{4xy}{\sqrt{x^2+4y^2}}
    nope


    2)
    z=ln(x^2+y^2)
    a) \frac{\partial^3 z}{\partial x \partial y^2}

    The answer is????


    a) \frac{4x^2+4y^2-8x^3-8xy^2}{(x^2+y^2)^3}
    nope

    3) z=\sqrt{2xy+y}
    a) \frac{ \partial^2 z}{ \partial x \partial y}

    b) \frac{ \partial^3 y}{ \partial x^3}

    How I do the 3?
    first write as z = (2xy + y)^{1/2}

    for (a), first differentiate with respect to x, then differentiate with respect to y

    for (b), take \frac {\partial z}{\partial x} and differentiate it with respect to x twice
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