Need help with Implicit differentiation for partial derivative

Question

I am currently taking Cal 3 and my professor taught in class that if I have a function $F(x,y,z)=c$ (where $c$ is a constant), then $\frac{dz}{dx}=-\frac{F_x}{F_z}$. I understand the textbook derivation on this formula, but I also tried to derive it myself by expanding $\frac{F_x}{F_z}=\frac{\frac{dF}{dx}}{\frac{dF}{dz}}=\frac{\frac{1}{dx}}{\frac{1}{dz}}=\frac{1}{dx} \cdot dz=\frac{dz}{dx}$. My derivation seems to be sound, but I know I'm missing something. I hope someone can show me where I went wrong and sorry for the messy notation as I'm very new to the site.

Answer 1

I think the confusion arises here because of notation.

Instead of $z$, let the implicit function be $(x,y) \mapsto g(x,y)$.

Then $F(x,y,g(x,y)) = c$ and differentiating gives ${\partial F(x,y,g(x,y)) \over \partial x} + {\partial F(x,y,g(x,y)) \over \partial z} {\partial g(x,y) \over \partial x} = 0$.

Reverting to your notation gives $F_x + F_z {dz \over dx} = 0$. (Note that the latter should really be ${\partial z \over \partial x}$.)