What is being differentiated in this example that uses the Euler-Lagrange equation? 
In this example of using the Euler-Lagrange equation, I am struggling to understand as to what the term being differentiate with respect to $x$ in line $(2.30)$ came from (it's not one of the partial derivatives of $F$ with respect to $y$ or $y'$). I am thinking that it might be from the equation: $$F_{y'y'}y'' = F_y - F_{y'y}y' - F_{y'x}$$
  where $F_{y'x} = 0$. But the term I am confused about isn't even equal to $F_{y'}$. 



For any magic there is a simple scientific explanation. In this case it is called Beltrami identity. Since the function $F$ does not depend explicitly on $x$, the Euler-Lagrange equation is equivalent (modulo to constants $y'=0$) to the following equation
$$
F-y'F_{y'}=C,
$$
which after the differentiation and simplification becomes
$$
\frac{y}{\sqrt{1+y'^2}}=C.
$$
Here we recognize our old fellow in the LHS.
