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:(adsbygoogle = window.adsbygoogle || []).push({});

We have [itex]g(\epsilon) = f(t) + \epsilon h(t)[/itex]. We have to compute [itex]\frac{\partial F(g(\epsilon))}{\partial \epsilon}[/itex]. This is supposed to be equal to [itex]\frac{\partial F(f)}{\partial f}h(t)[/itex] when [itex]\epsilon = 0[/itex]. 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 [itex]f(t)+\epsilon h(t)[/itex] as [itex]f+\Delta f[/itex] since [itex]h[/itex] can be any arbitrary function, and that allows me to use the other definition of the derivative. However, it does not seem like a very rigorous way of doing it.

How can I show that [itex]\frac{\partial F(g(0))}{\partial \epsilon} = \frac{\partial F(f)}{\partial f}h(t)[/itex] using the definition of the derivative? Or, rather, a definition of the derivative..?

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# I Rigorously understanding chain rule for sum of functions

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