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Homework Help: Need help on a partial derivative problem!

  1. Apr 27, 2005 #1

    jzq

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    Find the second-order partial derivatives of the given function. In each case, show that the mixed partial derivatives [tex]f_{xy}[/tex] and [tex]f_{yx}[/tex] are equal.

    Function:
    [tex]f(x,y)=x^{3}+x^{2}y+x+4[/tex]

    My work (Correct me if I am wrong):
    [tex]\frac{\partial{f}}{\partial{x}}}=3x^{2}+2xy+1[/tex]

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

    [tex]f_{xx}=6x+2y[/tex]

    [tex]f_{yy}=0[/tex]

    [tex]f_{xy}=6x+2y[/tex]

    [tex]f_{yx}=0[/tex]

    If I am correct, which I am probably not, how could [tex]f_{xy}[/tex] possibly be equal to [tex]f_{yx}[/tex]? Shouldn't that always be true anyways? If that's so, then obviously I messed up somewhere. Please help!
     
  2. jcsd
  3. Apr 27, 2005 #2
    How did you find those mixed partials? You seem to have done the exact same thing to find [itex]f_{xy}[/itex] as you did for [itex]f_{xx}[/itex] (and the same for [itex]yy[/itex] and [itex]yx[/itex]). I think if you check your work over, you'll see that you differentiated wrt the wrong variables a couple of times :wink:
     
  4. Apr 27, 2005 #3

    dextercioby

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    Use Jacobi's notation for partial derivatives.It will leave no room for any confusion once u realize the order of differentiation.And if u use Lagrange's one,do it properly

    [tex] \frac{\partial f}{\partial x}\equiv f'_{x} [/tex]

    Daniel.
     
  5. Apr 27, 2005 #4
    Nothing wrong with notation evolving. I've never seen notation like [itex]f^\prime_x[/itex], though.
     
  6. Apr 27, 2005 #5
    Did I atleast get the first partial derivatives correct?
     
  7. Apr 27, 2005 #6
    Your second partials are wrt to the wrong variables

    [tex] f_{xy} [/tex] means differentiate [itex] f_x [/itex] with respect to y.
     
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