So, I have an implicit expression for a change of variables from Cartesian to a different system using $u$ and $v$.
This is given by:
$G(x,y,z,u,v)=0$, $F(x,y,z,u,v)=0$, where $F$, $G$, $u=f(x,y,z)$ and $v=g(x,y,z)$ are all differentiable.
Differentiating F, say, we get:
$\frac{dF}{dx} = \frac{\partial{F}}{\partial{x}} + \frac{\partial{F}}{\partial{u}} \frac{\partial{u}}{\partial{x}} + \frac{\partial{F}}{\partial{v}}\frac{\partial{v}}{\partial{x}}$
My questions:
Is this because you have $\frac{\partial{x}}{\partial{x}} = 1$, and $\frac{\partial{y}}{\partial{x}} = \frac{\partial{z}}{\partial{x}} = 0$?
This seems like an obvious question but I don't really understand where the $ \frac{\partial{F}}{\partial{x}}$ term is coming from, or why is just appears here. For example if we had $f(x,y,z)=f(x(t), y(t), z(t))$, if we wanted to differentiate that with respect to t we wouldn't have an $ \frac{\partial{f}}{\partial{t}}$ term.
It's just the chain rule.
Perhaps it's easier to see if you rename things a little, so that you don't use the same symbols for the coordinates as for the functions involved.
If from the function $F(x,y,z,u,v)$ you form the composed function $$ f(r,s,t)=F\bigl(X(r,s,t),Y(r,s,t),Z(r,s,t),U(r,s,t),V(r,s,t)\bigr) , $$ then the chain rule says that $$ \begin{split} \frac{\partial f}{\partial r}(r,s,t) &= \frac{\partial F}{\partial x}\bigl(X(r,s,t),Y(r,s,t),Z(r,s,t),U(r,s,t),V(r,s,t)\bigr) \frac{\partial X}{\partial r}(r,s,t) \\ &+ \frac{\partial F}{\partial y}\bigl(X(r,s,t),Y(r,s,t),Z(r,s,t),U(r,s,t),V(r,s,t)\bigr) \frac{\partial Y}{\partial r}(r,s,t) \\ &+ \cdots \\ &+ \cdots \\ &+ \frac{\partial F}{\partial v}\bigl(X(r,s,t),Y(r,s,t),Z(r,s,t),U(r,s,t),V(r,s,t)\bigr) \frac{\partial V}{\partial r}(r,s,t) . \end{split} $$ In your case you have $$ X(r,s,t)=r ,\quad Y(r,s,t)=s ,\quad Z(r,s,t)=t $$ and $$ U(r,s,t)=f(r,s,t) ,\quad V(r,s,t)=g(r,s,t) . $$ Then $\partial X/\partial r=1$ and $\partial Y/\partial r=\partial Z/\partial r=0$, so you're left with $$ \begin{split} \frac{\partial f}{\partial r}(r,s,t) &= \frac{\partial F}{\partial x}\bigl(X(r,s,t),Y(r,s,t),Z(r,s,t),U(r,s,t),V(r,s,t)\bigr) \\ &+ \frac{\partial F}{\partial u}\bigl(X(r,s,t),Y(r,s,t),Z(r,s,t),U(r,s,t),V(r,s,t)\bigr) \frac{\partial f}{\partial r}(r,s,t) \\ &+ \frac{\partial F}{\partial v}\bigl(X(r,s,t),Y(r,s,t),Z(r,s,t),U(r,s,t),V(r,s,t)\bigr) \frac{\partial g}{\partial r}(r,s,t) , \end{split} $$ which is the formula that you asked about, in slightly different notation.