I'm have a bit of trouble understanding how dependence of variables work with implicit functions. This troubles me during the chain rule and specifically in this case, the implicit function theorem.

The way I have seen the IFT derived

Suppose we have a function $F(x,y,z)=c$, and we are finding $\frac{\partial F}{\partial x}$.

At this point, if I'm not mistaken, we asumme that $x$ and $y$ are "completely independent" and that $z$ can be expressed as a function of $x$ and $y$ locally. So $F(x,y,f(x,y))=c$

And then we differentate both sides with respect to $x$, using the chain rule on the left hand side $$\frac{\partial F}{\partial x}\cdot1+\frac{\partial F}{\partial y}\cdot0+\frac{\partial F}{\partial z}\cdot\frac{\partial z}{\partial x} =0$$

My question is regarding this:

Question. Why is it a reasonable assumption to make that $x$ and $y$ are completely independent? (Especially when they are related by the function $F(x,y,z)=c$) If this is false (i.e. $x,y$ are dependent), then how can we conclude $$ \frac{\partial y}{\partial x} = 0 $$ which seems to be required to derive the formula for the implicit function theorem?

My question also extends beyond just the implicit function theorem:

In general if we have a function $z=f(x(u),y(u))$ and we're differentiating with respect to $x$. Usually what I would do is $$ \frac{\partial z}{\partial x} = \frac{\partial f}{\partial x} $$ by treating $y$ as a constant.

But aren't $y$ and $x$ related? Why is $\frac{\partial y}{\partial x} = 0 $ if a change in $x$ suggests a change in $u$ and thus a change in $y$ (Since $y$ also depends on $u$)?


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