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Math Help - Isolating a Variable in Multivariable Function

  1. #1
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    Isolating a Variable in Multivariable Function

    This is actually for a Calculus question, but the Algebra part is what I'm stumped on.

    I'm given the equation:
    x^4 + y^4 + z^4 = 4xyz

    And I will need to find the partial derivatives of z with respect to x and y. That part I can do. But I haven't been able to isolate z.

    I feel like this should be relatively simple, but I'm missing something.
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  2. #2
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    Quote Originally Posted by TechVick View Post
    This is actually for a Calculus question, but the Algebra part is what I'm stumped on.

    I'm given the equation:
    x^4 + y^4 + z^4 = 4xyz

    And I will need to find the partial derivatives of z with respect to x and y. That part I can do. But I haven't been able to isolate z.

    I feel like this should be relatively simple, but I'm missing something.
    It is not easy to isolate z. That would mean solving a fourth degree equation.

    However, if you just want to find the derivative of z with respect to x, you don't need to solve for z. Use "implicit differentiation"- differentiate each part of the equation with respect to x, writing [tex]\frac{\partial z}{\partial x} for the unknown derivative:
    4x^3+ 0+ 4z^3\frac{\partial z}{\partial x}= 4yz+ 4xy\frac{\partial z}{\partial x}
    Now solve that linear equation for \frac{\partial z}{\partial x}.
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