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Finding the partial derivatives of function

  1. Nov 1, 2012 #1
    1. The problem statement, all variables and given/known data

    If [itex]z=\frac{1}{x}[f(x-y)+g(x+y)][/itex], prove that [itex]\frac{\partial }{\partial x}(x^2\frac{\partial z}{\partial x})=x^2\frac{\partial^2 z}{\partial y^2}[/itex]

    2. Relevant equations



    3. The attempt at a solution

    I don't know how I'm supposed to find the partial derivative of z with respect to any of variables if the function f and g are not expressed.
    Help me!

    Thank you
     
  2. jcsd
  3. Nov 1, 2012 #2

    LCKurtz

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    Just leave the f and g in there, maybe they will cancel out. For example, if you want the partial of f(x-y) with respect to y it would be:$$
    f_y(x-y) = -1\cdot f'(x-y)$$and so on.
     
  4. Nov 1, 2012 #3
    Can you explain me why [itex]f_y(x-y) = -1\cdot f'(x-y)[/itex]? I don't get it.
    What is then [itex]f_x(x+y) = ?[/itex]
     
  5. Nov 1, 2012 #4

    LCKurtz

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    It is the chain rule. If you have ##z=f(x-y)## let ##u = x-y## so ##z=f(u)##. Your chain rule for is$$
    z_x = z_u u_x,\ z_y = f_u u_y$$and since f depends on only one variable u, you would write ##f_u = f'(u)## and you multiply by the ##u_x## or ##u_y## accordingly.
     
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