The proof of the following Theorem is almost identical to the proof of the change of variables formula given in [@dvv:entropy2], together with the line of the proof of Theorem \[thrm: onechangeofvar\], and is therefore omitted.

\[thrm: genchangeofvar\]Let $ F_{i}\in B[t_{1},\dots ,t_{n}] $, $ i=1,\dots ,n $ be non-commutative power series with common multiradius of convergence $ (R_{1},\dots ,R_{n}) $. Assume that there are non-commutative power series $ G_{i} $, $ i=1,\dots ,n $ with common multiradius of convergence $ (r_{1},\dots ,r_{n}) $, such that for all $ i=1,\dots ,n $, $$\begin{aligned}
F_{i}(G_{1}(t_{1},\dots ,t_{n}),\dots ,G_{n}(t_{1},\dots ,t_{n}))=t_{i}, &  & \\
G_{i}(F_{1}(t_{1},\dots ,t_{n}),\dots ,F_{n}(t_{1},\dots ,t_{n}))=t_{i},. &  & \end{aligned}$$ Assume that for each $ i $, $ \Vert X_{i}\Vert <\min (r_{i},R_{i}) $. Then $$\begin{aligned}
\chi (F_{1}(X_{1},\dots ,X_{n}),\dots ,F_{n}(X_{1},\dots ,X_{n})|B)= &  & \\
\chi (X_{1},\dots ,X_{n}|B)+{{\operatorname{Tr}}}\otimes \tau \otimes \tau \left( \log |D_{B}F({X_{1},\dots ,X_{n}})|\right) . &  & \end{aligned}$$ The same formulas hold for $ \chi ^{\omega } $ in place of $ \chi  $.

We deduce that “free Brownian motion” has a regularizing effect on free entropy (compare [@dvv:entropy5]). The following proposition follows also from the results of [@dvv:entropy3], but we could not find its exact statement there.

Let $ S_{1},\dots ,S_{n} $ be a 