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Partial derivatives

  1. Apr 3, 2009 #1
    1. The problem statement, all variables and given/known data
    If z = 1 / (x^2+y^2-1)
    show that x(dz/dx)+y(dz/dy)=2z(1+z)

    2. The attempt at a solution
    z = (x^2+y^2-1)^-1
    dz/dx = -2x(x^2+y^2-1)^-2 = -2x * z^2
    dz/dy = -2y(x^2+y^2-1)^-2 = -2y * z^2

    (-2x^2 * z^2) - (2y^2 * z^2) = 2z(1+z)

    I can express x and y in something like z and x/y:
    x = z^-1 - (z^-1*y^2)
    y = z^-1 - (z^-1*x^2)

    Though substituting this values in the obtained equation doesn't get me near the answer...
  2. jcsd
  3. Apr 3, 2009 #2


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    Staff Emeritus
    Science Advisor

    Yes, exactly right. Why would you want to do the following?

  4. Apr 3, 2009 #3
    Because I have no idea how to get rid of the x and y...

    Thanks for your quick reply.
  5. Apr 3, 2009 #4
    There's a negative sign missing in the formula you need to show. It should be
    [tex] x \frac{dz}{dx} + y \frac{dz}{dy} = -2z(1+z)[/tex]
  6. Apr 4, 2009 #5
    It sounds like you're trying to get x and y in terms of z once you have found the partial derivatives. Leave the partials in terms of x and y, and then expand out the -2z(1+z) in terms of what z equals. This will help you verify the formula (please see my previous post for the correction to the problem statement).
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