I'm a Physics undergraduate, deeply enamored of math too. I'm trying to make a proof of Noether's theorem more precise with minimal abuse of notation as well, since almost in every physics resource, there's only sloppy notation which just sweeps most of the mathematical details under the rug. Anyways, I'm stuck on how to make the following step in the proof precise. Any help is appreciated.
Let $(\{Q_\varepsilon\}_{\varepsilon\in\mathbb{R}}, \circ)$ and $(\{T_\varepsilon\}_{\varepsilon\in\mathbb{R}}, \circ)$ be "smooth at $\varepsilon=0$" (see point 2 below) one-parameter groups homomorphic to $(\mathbb{R}, +)$, such that $Q_\varepsilon, T_\varepsilon : \mathbb{R\to R}$ are functions for each $\varepsilon\in\mathbb{R}$. Suppose that $\mathcal{F}$ is the function that does this: $\varepsilon\mapsto Q_\varepsilon\circ q\circ T_{-\varepsilon}$, where $q:\mathbb{R\to R}$ is a smooth function.
Question: What will the expression ${d\over d\varepsilon}\big|_{\varepsilon=0}(Q_\varepsilon\circ q\circ T_{-\varepsilon})$ evaluate to? (See point 3 below.)
Note:
Please restrain from using advanced concepts like tangent spaces and stuff (I don't know about this yet), unless you do not see any other approach (using simpler concepts). Nevertheless, if you have to, I'll really appreciate if you could give some naive explanation too.
By "smooth at $\varepsilon=0$", I mean (naively of course, sorry!) that all the "derivatives" $\bigl({d\over d\varepsilon}\bigr)^n\big|_{\varepsilon=0} Q_\varepsilon$ "make sense and exist"; and that we have the expansion $Q_\varepsilon = Q_0 + \varepsilon{d\over d\varepsilon}\big|_{\varepsilon = 0}Q_\varepsilon + O(\varepsilon^2)$ for small $\varepsilon$.
In point 2, I've abused notation (in my opinion). For instance, ${d\over d\varepsilon}\big|_{\varepsilon=0}Q_\varepsilon$ should be ${d\over d\varepsilon}\big|_{\varepsilon=0} \mathcal{Q}$ where $\mathcal{Q}$ is the function that does this: $\varepsilon\mapsto Q_\varepsilon$.