I've read in some lecture notes that the (vertical) infinitesimal variation of a field w.r.t a local parameter epsilon can be expressed as $$ \delta\phi^a = G^a\epsilon+G^{a,\mu}d_\mu\epsilon\tag{1}$$ for some (epsilon independent) functions $$G^a\hspace{0.2cm}and\hspace{0.2cm} G^{a,\mu} \tag{2}$$
I also know that the said variation can be expressed in terms of the Taylor expansion of the infinitesimal transformed field $$\delta\phi^a = {\phi^a}'(x)-\phi^a(x) = \epsilon\frac{\partial\phi^a}{\partial\epsilon} + \mathcal O(\epsilon^2)\tag{3}$$
My question is what is the formula for the $G$ functions? I am especially confused about the second one (how can a term involving the spacetime derivative of the parameter appear in the expansion)?
