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Chain Rule with Partial Derivative?

  1. Mar 16, 2015 #1
    1. The problem statement, all variables and given/known data
    Given that the surface x^7y^2+y^4z^6+z^8x^8+9xyz=12 has the equation z=f(xy) in a neighbourhod of the point (1,1,1) with f(x,y) differentiable, find the derivatives.

    df/dx (1,1) = ???

    d^2f/dx^2 (1,1) = ???

    2. Relevant equations


    3. The attempt at a solution
    df/dx (1,1) I got -24/23 or
    7x^6y^2+6y^4z^5(dz/dx)+8z^7x^8(dz/dx)+8z^8x^7+9yz+9xy(dz/dx)

    d^2f/dx^2: I tried doing the same thing and squaring the derivative when they came up again but it didn't work.
     
  2. jcsd
  3. Mar 16, 2015 #2

    SteamKing

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    When you take the partial derivative of f(x,y,z) w.r.t. x, you treat any y's or z's as constants.

    For example, if f(x,y,z) = 3x2y + y2z, then

    ∂ f / ∂ x = 6xy

    The chain rule would be used only if you know y and/or z as functions of x.
     
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