Second-order partial derivatives of f(x,y)=x³+x²y+x+4

[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.