I am currently reading about Calculus of Variations, and I have two questions:
The author introduces the Euler-Lagrange equation $F_{y'y'}y'' = F_y - F_{y'y}y' - F_{y'x}$. The final step in which the author derives it, he uses the chain rule to calculate
$\frac{d}{dx}(\frac{\partial F}{\partial y'}) = \frac{\partial^2 F}{\partial y'\partial x} + \frac{\partial^2 F}{\partial y'\partial y}\cdot y' + \frac{\partial^2 F}{\partial y'^2}\cdot y''$.
Perhaps this is just some elementary derivations going on, but I don't understand how one would go about doing taking the derivative of a partial derivative w.r.t a dependent variables derivative (I think I got that right). I.e. I have no clue what is going on here. Any clarily here would be appreciated.
He then gives an example of a function $F(x, y, y') = y\sqrt{1 + (y')^2}$, and says that $F_{y'x} = 0$ since the function is independent of $x$. But $y$ is a function of $x$, so this isn't exactly true. Right?
Am I wrong? And if so, what could be the root of my misunderstanding here?