in $t$. Corollary \[crll:exponential\_inO\] asserts that the remainder sum is in ${\mathcal{O}}\left(\alpha^{n}\Gamma(n+\beta-R)\right)$.

Proof of the main theorem: Composition of power series in ${{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$
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Despite the fact that Bender’s theorem applies to a broader range of compositions $f \circ g$, where $f$ does not need to be analytic and $g$ does not need to be an element of ${{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$, it cannot be used in the case $f,g \in {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$, where $f \notin \ker {\mathcal{A}}^\alpha_\beta$. The problem is that we cannot truncate the sum $\sum_{k=0}^\infty f_k g(x)^k$ without losing significant information. In this section, this obstacle will be confronted and the general chain rule for the asymptotic derivative will be proven. Let $\operatorname{Diff}_{\operatorname{id}}({\mathbb{R}},0):= ( \{f \in {\mathbb{R}}[[x]]: f_0=0 \text{ and } f_1=1 \}, \circ)$ denote the group of *formal diffeomorphisms tangent to the identity*. It is the group of all power series with $f_0=0$ and $f_1=1$ and with composition as group operation. The restriction of this group to elements in ${{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ is of special interest to us.

\[thm:chainrule\] $\operatorname{Diff}_{\operatorname{id}}({\mathbb{R}},0)^\alpha_\beta:= ( \{f \in {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}: f_0=0 \text{ and } f_1=1 \}, \circ)$ is a subgroup[^2] of $\operatorname{Diff}_{\operatorname{id}}({\mathbb{R}},0)$. Moreover, ${\mathcal{A}}^\alpha_\beta$ fulfills a chain rule: If $f,g \in {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ with $g_0=0,g_1=1$, then $f \circ g, 