  We combine Kirchheim's metric differentials with Cheeger charts in order to
establish a non-embeddability principle for any collection $\mathcal C$ of
Banach (or metric) spaces: if a metric measure space $X$ bi-Lipschitz embeds in
some element in $\mathcal C$, and if every Lipschitz map $X\to Y\in \mathcal C$
is differentiable, then $X$ is rectifiable. This gives a simple proof of the
rectifiability of Lipschitz differentiability spaces that are bi-Lipschitz
embeddable in Euclidean space, due to Kell--Mondino. Our principle also implies
a converse to Kirchheim's theorem: if all Lipschitz maps from a domain space to
arbitrary targets are metrically differentiable, the domain is rectifiable. We
moreover establish the compatibility of metric and w$^*$-differentials of maps
from metric spaces in the spirit of Ambrosio--Kirchheim.
