In my quest to understand the Euler-Lagrange equation, I've realized I have to understand the chain rule first. So, here's the issue:
We have $g(\epsilon) = f(t) + \epsilon h(t)$. We have to compute $\frac{\partial F(g(\epsilon))}{\partial \epsilon}$. This is supposed to be equal to $\frac{\partial F(f)}{\partial f}h(t)$ when $\epsilon = 0$. However, this does not make any sense to me. Doing the computations and using the chain rule, I get: $$\frac{\partial F(g(0))}{\partial \epsilon} = \lim_{\epsilon \to 0}\frac{F(g(\epsilon)) - F(g(0))}{g(\epsilon) -g(0) } \frac{g(\epsilon) -g(0)}{\epsilon} = \lim_{\epsilon \to 0}\frac{F(f(t)+\epsilon h(t)) - F(f(t))}{\epsilon h(t) } h(t) $$ On an intuitive level I can understand it. I can think of $f(t)+\epsilon h(t)$ as $f+\Delta f$ since $h$ can be any arbitrary function, and that allows me to use the other definition of the derivative. However, I do not think this is a very rigorous way of doing it.
How can I show that $\frac{\partial F(g(0))}{\partial \epsilon} = \frac{\partial F(f)}{\partial f}h(t)$ using the definition of the derivative? Or, rather, a definition of the derivative..?