What does it mean to take $F(x, y)$ as $F(x, y(x))$? I just do, not, get the difference, despite numerous explanations.
To put it into context, I see this done when applying the chain rule to $F$ to find $\frac{\mathrm{d}F}{\mathrm{d}x}$.
So what does it mean? What is the difference between $F(x, y)$ and $F(x, y(x))$?
i.e. take $F(x, y) = x^2 + y^3 + 2y$. What would $F(x, y(x))$ be equal to?
Please dumb explanations down, and don't skip steps, as I've heard numerous explanations and don't get any of them!
$y(x)$ seems to denote a function in the variable $x$. So if $F(x,y)$ is given for $(x,y) \in \mathbb{R}^2$, $F(x,y(x))$ means the evaluation of $F$ at $(x,y(x))$ : you get a function for the variable $x$.
Your confusion must be the double use of $y$ as a variable and as a function. Rather, call $\varphi(x)$ instead of $y(x)$. Then you are actually just looking at $F(x,\varphi(x))=x^2+\varphi(x)^3+2\varphi(x)$ (you replace the variable $y$ by the value $\varphi(x)$) and the chain rule explain how to get the derivative of $x \mapsto F(x,\varphi(x))$ in terms of the partial derivatives of $(x,y) \mapsto F(x,y)$