According to this wikipedia, in section Euler–Lagrange equation, in the first example (shortest path between two points), it says
$${\displaystyle {\frac {\partial L}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}=0}$$
with
$${\displaystyle L={\sqrt {1+[f'(x)]^{2}}}\,.}$$
Since f does not appear explicitly in L , the first term in the Euler–Lagrange equation vanishes for all f (x) and thus,
$${\displaystyle {\frac {d}{dx}}{\frac {\partial L}{\partial f'}}=0\,.}$$
..
I am confused. Are f and f' independent so that the first term in the Euler–Lagrange equation vanishes for all f (x)?
In short, why is $${\displaystyle {\frac {\partial L}{\partial f}}=0}$$ where $${\displaystyle L={\sqrt {1+[f'(x)]^{2}}}\,.}$$ ?
As $${\displaystyle {\frac {\partial L}{\partial f}}={\frac {\partial f'}{\partial f}}×{\frac {\partial L}{\partial f'}}}$$, does this mean that $${\displaystyle {\frac {\partial f}{\partial f'}}=0}$$ ?
Thank you in advance for making it clear for me.
