Partial Differentiation

[FizzyCrow]

I do not have a specific problem in mind, i just need to be told, in stupid terms hehehe, how to do a second partial differentiation. I know that there is the partial one with respect to x^2 and y^2, i an having problems understanding how you get the xy, yx one. I also know that they are, in my case, always equal. Any help is appreciated, if you need a problem, i can see if i can pull one from my book.


I also wanted to say thank you to everyone that helped me in my last posts, i've just been so busy, i have not been able to respond, so THANKS

 

[Unco]

$\displaystyle f(x,y) = x^2 + y^2$

$\displaystyle f_x$ means the partial derivative of f(x,y) with respect to x. As you know, we treat y as a constant and differentiate wrt x:

$\displaystyle f_x = 2x$
or $\displaystyle \frac{\partial f}{\partial x} = 2x$

Likewise,
$\displaystyle f_y = 2y$ or $\displaystyle \frac{\partial f}{\partial y} = 2y$

$\displaystyle f_{xy}$ reads from left to right in that: "the partial derivative of $\displaystyle f_x$ wrt y".

But with the chain rule reads:

$\displaystyle \frac{\partial}{\partial y} \left(\frac{\partial f}{\partial x}\right)$ (which here = 0)