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 f_{xy} and f_{yx} are equal.

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

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

\frac{\partial{f}}{\partial{y}}}=x^{2}

f_{xx}=6x+2y

f_{yy}=0

f_{xy}=6x+2y

f_{yx}=0

If I am correct, which I am probably not, how could f_{xy} possibly be equal to f_{yx}? 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 f_{xy} as you did for f_{xx} (and the same for yy and yx). 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

\frac{\partial f}{\partial x}\equiv f'_{x}

Daniel.

 

[Data]

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

 

[ddluu]

Did I atleast get the first partial derivatives correct?

 

[whozum]

Your second partials are wrt to the wrong variables

f_{xy} means differentiate f_x with respect to y.