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Math Help - Implicit Differentiation of Partial Derivatives

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    Implicit Differentiation of Partial Derivatives

    Question:  ln(x^5z^{216}) + 9 (x - 1)y + 216xz + 64y^3z = 0 defines y implicitly as a function of x and z; that is  y =y(x,z) . Compute  \frac{\partial y}{\partial z}  at (x,z) = (1,-1) .
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Belowzero78 View Post
    Question:  ln(x^5z^{216}) + 9 (x - 1)y + 216xz + 64y^3z = 0 defines y implicitly as a function of x and z; that is  y =y(x,z) . Compute  \frac{\partial y}{\partial z}  at (x,z) = (1,-1) .
    How much have you tried?

    \frac{\partial}{\partial z}\left[\ln(x^5z^{216})\right] should be routine.

    \frac{\partial }{\partial z}\left[9(x-1)y\right]=9(x-1)\frac{\partial y}{\partial z}

    \frac{\partial}{\partial z}\left[216xz\right] should be routine.

    \frac{\partial}{\partial z}\left[64y^3z\right] requires product rule.

    Combine all these results, and then solve for \frac{\partial y}{\partial z}

    Can you take it from here?
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    Quote Originally Posted by Chris L T521 View Post
    How much have you tried?

    \frac{\partial}{\partial z}\left[\ln(x^5z^{216})\right] should be routine.

    \frac{\partial }{\partial z}\left[9(x-1)y\right]=9(x-1)\frac{\partial y}{\partial z}

    \frac{\partial}{\partial z}\left[216xz\right] should be routine.

    \frac{\partial}{\partial z}\left[64y^3z\right] requires product rule.

    Combine all these results, and then solve for \frac{\partial y}{\partial z}

    Can you take it from here?
    Could please show me what happens with the y in the last two terms? When i evaluate the derivative wont I have a problem?
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    In the last two terms? There is no y in 216xz.

    \frac{\partial 64y^3z}{\partial x}= 64(3y^2)\frac{\partial y}{\partial x} z= 192y^2z \frac{\partial y}{\partial x}.

    \frac{\partial 64y^3z}{\partial z}= 64(3y^2)\frac{\partial y}{\partial z} z+ 64y^3=  192y^2z\frac{\partial y}{\partial z}+ 64y^3.
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    Here is what i got. Can anyone please confirm?

     \frac{\partial y}{\partial z} = \frac{\frac{216}{z} - 216x -64y^3}{9x -9 + 192zy^2}

    I am confused onto find what z is. Could someone please explain. I know that y is a function of x and z, but how do i find it?

    Thanks!

    Nevermind, i found  y = \frac{3}{2} . I's unsure of why the answer is -1/2. Im missing a negative sign somewhere.... since im getting 1/2.
    Last edited by Belowzero78; April 13th 2010 at 02:04 PM.
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  6. #6
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    SOLVED. I figured out my mistakes... thanks everybody.
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