My calc textbook says that if you have a function F such that F(x, f(x) = y) = $0$, then you can differentiate both sides of the equation with respect to x to receive the following:
$$\frac{dF}{dx}=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y}\frac{dy}{dx}=0$$
And, therefore, assuming $\frac{\partial F}{\partial y}\neq 0 $:
$$\frac{dy}{dx}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}= -\frac{F_x}{F_y}$$
Here is my confusion. F would simply be a two-dimensional level surface. Since the value of F is constant (zero), wouldn't $F_y$ necessarily be $0$, as well as $F_x?$ After all, the rate of change of the height of a level surface in any direction should be $0$, right? This issue reminds me of an idea from calc $1$:
$$x = 0$$ $$ \frac{d}{dx}[x] = \frac{d}{dx}[0]$$ $$1 = 0$$
Of course, this assumption is false because it doesn't take into account that x is a constant. In this case, $\frac{d}{dx}[x] =0$. So why doesn't this also apply here with $F_x$ and $F_y$?