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Partial derivatives of function log(x^2+y^2)

  1. Jun 20, 2012 #1
    1. The problem statement, all variables and given/known data
    I have got a question concerning the following function:

    [tex]f(x,y)=\log\left(x^2+y^2\right)[/tex]​

    Partial derivatives are:

    [tex]\frac{\partial^2f}{\partial x^2}=\frac{y^2-x^2}{\left(x^2+y^2\right)^2}[/tex]​

    and

    [tex]\frac{\partial^2f}{\partial y^2}=\frac{x^2-y^2}{\left(x^2+y^2\right)^2}[/tex]​

    The conclusion is that the following equation is right:

    [tex]\frac{\partial^2f}{\partial x^2}=-\frac{\partial^2f}{\partial y^2}[/tex]​

    But I can not understand, how can it be possible. The role of x and y variables are exactly the same, then why derivatives are not the same?

    Sorry for my English - it is my second language. I am from Poland.
     
  2. jcsd
  3. Jun 20, 2012 #2

    Curious3141

    User Avatar
    Homework Helper

    There's a factor of 2 missing in all your second derivatives.

    The result is exactly as you'd expect. The variable you're differentiating with respect to, matters. If it's x, then y is treated as a constant, and vice versa. So if the "active" variable is leading in the numerator in one derivative, the same should apply in the other. It's just that the "active" variable is x in one case and y in the other, and the other variable acts like a constant.
     
  4. Jun 21, 2012 #3
    How did you get those partial derivatives? They are wrong.


    P.S. There's nothing wrong with your English, and even if there were, there is nothing to apologise for.
     
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