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Let $G_{M}^{+}$ be the Artin monoid of finite type generated by the letters$a_i, i\in I$ with respect to a Coxeter matrix $M$ that is equipped with thedegree map $\deg\!:\!G_{M}^{+} \to\!\Z_{\ge0}$ defined by assigning to eachequivalence class of words the length of the words, and let $N_{M,\deg}(t)\!:=\!\!\sum_{J \subset I}(-1)^{\#J} t^{\deg(\Delta_{J})}$ be theskew-growth function, where the summation index $J$ runs over all subsets of$I$ and $\Delta_{J}$ is the fundamental element in $G_{M}^{+}$ associated tothe set $J$. In this article, we will calculate the derivative at $t = 1$ ofthe polynomial $N_{M, \deg}(t)$. As a result, we show that the polynomial$N_{M, \deg}(t)$ has a simple root at $t = 1$.

On the derivative at $t = 1$ of the skew-growth functions for Artin monoids