Need help on a partial derivative problem

[jzq]

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!

 

[Data]

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

 

[dextercioby]

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.

 

[Data]

Nothing wrong with notation evolving. I've never seen notation like [itex]f^\prime_x[/itex], though.

 

[ddluu]

Did I atleast get the first partial derivatives correct?

 

[whozum]

Your second partials are wrt to the wrong variables

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