gives eq.  with the substitution $g \rightarrow g^{-1}$.

Bender and Richmond [@Bender1984] established that $[x^n](1+g(x))^{\gamma n + \delta} = n \gamma e^{\frac{\gamma g_1}{\alpha}} g_n + {\mathcal{O}}(g_n)$ if $g_n \sim \alpha n g_{n-1}$ and $g_0 = 0$. Using Lagrange inversion the first coefficient in the expansion of the compositional inverse in eq.  can be obtained from this. In this way, Theorem \[thm:chainrule\] is a generalization of Bender and Richmond’s result. In the same article Bender and Richmond proved a theorem similar to Theorem \[thm:chainrule\] for the class of power series $f$ which grow more rapidly than factorial such that $n f_{n-1} \in {o}(f_n)$. Theorem \[thm:chainrule\] establishes a link to the excluded case $n f_{n-1} = {\mathcal{O}}(f_n)$.

The chain rule in eq.  exposes a peculiar algebraic structure. It would be useful to have a combinatorial interpretation of the $e^{\frac{g(x) - x}{\alpha x g(x)}}$ term.

Applications {#sec:applications}
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Connected chord diagrams
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A chord diagram with $n$-chords is a circle with $2n$ points, which are labeled by integers $1,\ldots,2n$ and connected in disjoint pairs by $n$-chords. There are $(2n-1)!!$ of such diagrams. A chord diagram is *connected* if no set of chords can be separated from the remaining chords by a line which 