Distinguish dF / dx and ∂F / ∂x in "Implicit function theorem"

Question

I have a question about derivation to distinguish the symbols dF / dx and ∂F / ∂x in the following equation: dF / dx = 0 = ∂F / ∂x + (∂F / ∂z) * (∂z / ∂x) , I cannot understand what distinguishes the complete derivative of dF / dx and the partial derivative ∂F / ∂x.

Answer 1

What you have is $$F = F(x,z) = F(x,z(x))$$ So $F$ is a function depending on 2 variables, with the second variable being a function itself which depends on the first variable. So when you compute $$\frac{d}{dx}[F(x,z(x))] = \frac{\partial F}{\partial x}(x,z(x)) + \frac{\partial F}{\partial z}(x,z(x))\frac{\partial z}{\partial x}(x)$$ The partial derivative with respect to $x$ means just the first variable $x$ is changing while $z(x)$ remains constant. For the total derivative we consider how the whole function $F(x,z(x))$ changes with $x$.

More precisely: $$\frac{d}{dx}[F(x,z(x))] = \lim_{h\rightarrow 0} \frac{F(x+h,z(x+h)) - F(x,z(x))}{h}$$ while $$\frac{\partial F}{\partial x}(x,z(x)) = \lim_{h\rightarrow 0}\frac{F(x+h,z(x)) - F(x,z(x))}{h}$$ and $$\frac{\partial F}{\partial z}(x,z(x)) = \lim_{h\rightarrow 0}\frac{F(x,z(x) + h) - F(x,z(x))}{h}$$