I have been working on the euler lagrange equations for physics, and they very briefly discussed the build up of it. I want to know more about it so I started to use wiki for help but I am stuck on this step (below) wiki
\begin{equation} \frac{dJ_\varepsilon}{d\varepsilon} = \int_a^b\left[\eta(x)\frac{\partial L_\varepsilon}{\partial f_\varepsilon} +\eta'(x) \frac{\partial L_\varepsilon}{\partial f'_\varepsilon}\right] dx \end{equation}
When $\varepsilon = 0$, then $g_\varepsilon = f$ and (etc).<<< I understand this.
A step inbetween could be: \begin{equation} \left.\frac{dJ_\varepsilon}{d\varepsilon}\right|_{\varepsilon = 0} = \int_a^b\left.\left[\eta(x)\frac{\partial L_\varepsilon}{\partial f_\varepsilon} +\eta'(x) \frac{\partial L_\varepsilon}{\partial f'_\varepsilon}\right]\right|_{\varepsilon=0} dx =0 \end{equation}
Then (need help on this step)
\begin{equation} \left.\frac{dJ_\varepsilon}{d\varepsilon}\right|_{\varepsilon = 0} = \int_a^b\left[\eta(x)\frac{\partial L}{\partial g} +\eta'(x) \frac{\partial L}{\partial g'}\right] dx = 0 \end{equation}
My question is, How did we apply the $\varepsilon = 0$ before actually solving this $\left[\eta(x)\frac{\partial L_\varepsilon}{\partial f_\varepsilon} +\eta'(x) \frac{\partial L_\varepsilon}{\partial f'_\varepsilon}\right]$ ?
Example:
let $y(x,\alpha) =x + \alpha $
then $k_1 = \frac{\partial y}{\partial \alpha} = 1$ now setting $\alpha = 0$, $k = 1$
but letting $\alpha = 0$ first, we get $y(x,0) =x$
then $k_2 = \frac{\partial y}{\partial \alpha} = 0$
thus $k_1 \neq k_2$ and so order matters
So clearly I am not understanding this. I tried looking for more information on this step (leibniz integral rule here , page 21), but its still not clear.
I would be very greatful if you could expand apon the step.