1-parametric subgroups of diffeomorphisms induce a complete vector field

Question

I have been working through this book on differential equations and I do not quite understand the justification for one claim. Namely, the author claims that every 1-parameter subgroup $\{\psi_\epsilon|\epsilon\in\mathbb{R}\}$ of diffeomorphisms gives rise to a vector field given by $$ V(x) = \left.\frac{d}{d\epsilon}\right|_{\epsilon = 0}\psi_\epsilon(x).$$ I am having trouble seeing the justification for the use of derivative. Is every 1-parametric subgroup automatically smoothly dependent on the parameter? Or is there a smoothness assumption I am missing?