Are all arguments held constants in partial derivatives? $F$ be a function of $x,$ $y,$ $z.$ $z$ depends on $x$ and $y.$ Then when evaluating $\partial F/\partial x,$ do I treat $z$ as a constant? I am under the impression that if I don't, what I would get is the total derivative $dF/dx.$ 
 A: I think you are saying , given a function of three variables $G(x,y,z)$ and another function of two variables $g(x,y)$ we define a function of two variables $F(x,y)=G(x,y,g(x,y))$, yes?  
Assume that is what you meant. Then, what is $\frac {\partial F}{\partial x}$?  Well, it is $$\frac {\partial F}{\partial x}=\lim_{h\to 0}\frac {F(x+h,y)-F(x,y)}{h}=\lim_{h\to 0}\frac {G(x+h,y,g(x+h,y))-G(x,y,g(x,y))}h$$
So you can see that the $z-$component of $G$ is not constant.  The derivative will involve both $\frac {\partial G}{\partial x}$ and $\frac {\partial G}{\partial z}$.  Indeed, it is not difficult to show from this that $$\frac {\partial F}{\partial x}=\frac {\partial G}{\partial x}+\frac {\partial G}{\partial z}\frac {\partial g}{\partial x}$$
Example:  say $G(x,y,z)=x+y +z^2$ and let $g(x,y)=2x+3y$.  Then define $$F(x,y)=G(x,y,g(x,y))=x+y+(2x+3y)^2=4x^2+x+y+12xy +9y^2$$
Of course we have $$\frac {\partial F}{\partial x}=8x+1+12y$$
We have the partials for $G$, namely $$\frac {\partial G}{\partial x}=1\quad \frac{\partial G}{\partial y}=1 \quad \frac {\partial G}{\partial z}=2z$$
We claim that $$\frac {\partial F}{\partial x}=\frac {\partial G}{\partial x}+\frac {\partial G}{\partial z}\frac {\partial g}{\partial x}$$
But the left hand is $$1+ 2z\times 2=1+4(2x+3y)=8x+12y+1$$  which is consistent.
A: You need more careful notation. You start with a function $F(x,y,z)$ and you form a new function $\phi(x,y) = F(x,y,g(x,y))$. Do you mean $\partial F/\partial x$ or do you mean $\partial\phi/\partial x$? [The latter is what you're referring to as the total derivative.] As it stands, your question really is ill-defined, but given the way you phrased it, I would vote unequivocally for the latter; otherwise, you must define the function $F$ with three independent variables.
