How do you compute a total differential when two ore more variables are changing simultaneously, and their partial derivative depend on each other. How do you compute a total differential when two ore more variables are changing simultaneously, and their partial derivative depend on each other.
Say I have $y = f(x_1, x_2, x_3, x_4)$
and 
$dy/dx_1$ depends on $x_2$,
$dy/dx_2$ depends on $x_1$,
and both $x_1$ and $x_2$ are changing at the same time.
In particular I want to distill the part of the change on y that is explained by the change in $x_1$, and the part that is explained by $x_2$.
I computed the total differential as it's done in any calculus textbook:
$$dy = f'_{x_1}dx_1 + f'_{x_2}dx_2 + f'_{x_3}dx_3 + f'_{x_4}dx_4$$
But what I get doesn't add up to the change I see in $y$.
I figured it's because, as I said, the partial derivatives of $x_1$ and $x_2$ depend on the other variable, but I don't really know how to deal with this... Maybe it's really simple, but I've been thinking it out for a couple of hours and I can't come up with a solution.
 A: The simplest example of the phenomenon you describe is to take
$$f(x,y) = xy,$$
so $\dfrac{\partial f}{\partial x} = y$ and $\dfrac{\partial f}{\partial y} = x$. Suppose you start at $(x,y)=(a,b)$ and make small changes $\Delta x$ and $\Delta y$ to $x$ and $y$ respectively. If you fix $y=b$ and vary $x$, then the area of the rectangle changes precisely by the amount $b\Delta x$ (which is $\dfrac{\partial f}{\partial x}(a,b)\Delta x$. If you fix $x=a$ and vary $y$, the area changes, analogously, by $\dfrac{\partial f}{\partial y}(a,b)\Delta y$. [In terms of the total change, if you draw the picture, you see that you do get the net change in $f(x,y)$ "to first order": You're missing only the tiny rectangle with area $\Delta x\Delta y$.]
What seems to be confusing you is that the rate of change of $f$ with respect to $x$ may, in fact, be $y$. Indeed, only rarely does $\dfrac{\partial f}{\partial x}$ depend only on $x$. You're misinterpreting what the partial derivative means if you expect that to be the case.
