The Chain Rule for Free Difference Quotients under an Affine Substitution

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $m,n\in\mathbb{N}$, let $T=(A,c)$ be an affine datum from $m$ to $n$ variables with affine substitution $\sigma_{T}:\mathcal{P}_{n}\to\mathcal{P}_{m}$, let $\gamma\in\Sigma_{m}$, and let $\mu=\gamma\circ\sigma_{T}$, which belongs to $\Sigma_{n}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint. Let $\partial^{\gamma}_{k}$ and $\partial^{\mu}_{i}$ be the free difference quotients of $\gamma$ and $\mu$.

¶For every $p\in\mathcal{P}_{n}$ and every $k\in[m]$,

$$\partial^{\gamma}_{k}\bigl(\sigma_{T}(p)\bigr)=\sum_{i=1}^{n}A_{ik}\,\partial^{\mu}_{i}(p).$$