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Complicated implicit multivariable differentiation problem

  1. Nov 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Given that the surface [itex]x^{6}y^{5}+y^{4}z^{5}+z^{9}x^{7}+4xyz=7[/itex] has the equation z = f(x, y) in a neighborhood of the point (1, 1, 1) with f(x,y) differentiable, find:

    [itex]\displaystyle\frac{\partial^{2} f}{\partial x^{2}}(1,1) = ? [/itex]

    2. Relevant equations



    3. The attempt at a solution

    To make things easier, i have already found an expression for the partial derivative of z with respect to x:

    [itex]\displaystyle\frac{\partial f}{\partial x} = \displaystyle\frac{-6x^{5}y^{5}-7z^{9}x^{6}-4yz}{5y^{4}z^{4}+9z^{8}x^{7}+4xy} [/itex]

    And at (1, 1), it's value is -17/18. I have tried to differentiate the expression with respect to x going from this general expression and doing so implicitly and then collecting the terms, however, i get two different results which are both wrong: 1129/729 and -160416. Is there an easier way to approach this problem or it is just tedious differentiation and being extremely careful with the terms?
     
    Last edited: Nov 18, 2013
  2. jcsd
  3. Nov 18, 2013 #2

    danago

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    Gold Member

    Should that first term on the denominator be 5y4z4 instead?
     
  4. Nov 18, 2013 #3
    Yes i'm sorry. I've fixed it but it won't affect the result anyways (as we have a y = 1 and we are differentiating with respect to x)
     
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