How to compute partial derivatives of $f(x(u,v), y(u,v))$ with respect to intermediate functions? I was going over the answers in this question and was wondering: 


*

*In the case of a function defined by $f(x(u,v), y(u,v))$, are $\partial f/\partial x$ and $\partial f/\partial y$ computed in the "normal" way where we just treat $x$ and $y$ as regular variables and not functions? 

*What happens if $x(u,v)$ and $y(u,v)$ are such that $y$ can be expressed in terms of $x$ aka $y=y(x)$? In this case is $\frac{\partial f(x,y(x))}{\partial y}$ the same as in the above where $x$ is constant or is $\frac{\partial f(x,y(x))}{\partial y}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial y}+\frac{\partial f}{\partial y}$ since if $y=y(x)$ then it's possible $x=x(y)$? Similarly, is $\frac{\partial f(x,y(x))}{\partial x}=\frac{\partial f}{\partial y}\frac{\partial y}{\partial x}+\frac{\partial f}{\partial x}$ or just $\frac{\partial f}{\partial x}$?   


In general, how do I interpret $\partial f/\partial x$ and $\partial f/\partial y$ in either case, given that you likely can't alter $x$ without altering $y$ and vice versa? 
 A: *

*Yes, $\partial f/ \partial x$ measures how $f$ changes when we change its first argument $x$. In this case, the first argument happens to be a function of two other quantities $u$ and $v$. 

*Suppose $y = y(x)$ is a function of $x$. Then by (1) the $y$-partial $\partial f/ \partial y$ is taken "normally" as if $y$ is an independent variable. However, when computing $\partial f/ \partial x$ we have to use the chain rule, since changing $x$ causes both of $f$'s arguments to change. 
The $x$-partial works out to 
$$ \frac{\partial f(x, y(x))}{\partial x} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial x} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial x} = \frac{\partial f}{\partial x} \cdot 1 + \frac{\partial f}{\partial y}\frac{\partial y}{\partial x}.$$ 
To summarize: the quantities $\partial f / \partial x$ and $\partial f/ \partial y$ measure how $f$ changes under small perturbations of its first and second argument, respectively. The product term in the chain rule accounts for when the first argument inadvertently changes the second argument. 
