family of embeddings between disks in $\mathbb{C}$ parametrized by a finite dimensional vector space $\{a\}$ and $u$ is a map from a fixed disk to ${\mathbb{R}}^{2n}$ of class $H^{3,\delta}$. Observe that if we restrict to a subspace $E_1\subset E_0$ of pairs $(u,a)$ where the map $u$ in the first entry enjoys one higher regularity (here $E_1$ is still considered as a Banach space, a subspace of $E_0$ singled out by a stronger norm), then $f: E_1\to F_0$ is $C^1$. Then for $x\in E_1$, the Fréchet derivative $df(x): E_1\to F_0$ drops regularity levels, which prevents composition of differentials, let alone the chain rule. But if we assume $E_1$ is dense in $E_0$ and $df(x)$ extends to $Df(x): E_0\to F_0$ so that $Df(x)$ is bounded, then we can still hope for the chain rule. If we try to establish the chain rule for the composition of two maps of these features, we really need the inclusion $E_1\subset E_0$ to be also compact, similar requirement for a $F_1$ of $F_0$ with $f(E_1)\subset F_1$ (as $F_1$ is the base domain of another tangent map to compose) as well as the continuity of $(x,v)\mapsto Df(x)v$, $E_1\times E_0\to F_0$. We will asume all that for this 