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Chain Rule Problem (Partial derivatives)

  1. Aug 10, 2015 #1
    1. The problem statement, all variables and given/known data
    pic.PNG

    2. Relevant equations


    3. The attempt at a solution
    I have the solution to this problem and the issue I'm having is that I don't understand this step:

    pic2.PNG

    Maybe I'm overlooking something simple but, for the red circled part, it seems to say that ∂/∂x(∂z/∂u) = (∂2z/∂u2)y+(∂2z/∂v∂u)(-y/x2). I realize that ∂2z/∂x∂u = ∂2z/∂u∂x but I can't see how either of them leads to the above. I almost understand it when I try to take the latter but apparently they take y and y/x^2 as constants and I don't understand why that is (assuming that is indeed what the solution is doing).
     
  2. jcsd
  3. Aug 10, 2015 #2

    Orodruin

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    This is generally not true. The partial derivative with respect to x implicitly assumes y to be constant and the partial derivative with respect to u assumes v to be constant.

    You are looking to apply the chain rule in its most basic form:
    $$
    \partial_x = \frac{\partial u}{\partial x} \partial_u + \frac{\partial v}{\partial x} \partial_v.
    $$
    What is being used is simply the very first line, but with ##z## replaced by ##\partial z/\partial u##.
     
  4. Aug 10, 2015 #3
    Oh of course, I remember how to do this now. ∂z/∂u is a function of (u,v) which are functions of (x,y) so I just apply the chain rule like usual.

    I made some really bad mistakes here (especially applying Clairaut's theorem so incorrectly) but at least the problem now looks pretty straightforward. Thanks for the help.
     
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