We develop an intrinsic linearized framework for metric differentiability of mappings from Euclidean domains into a metric space $X$.
Using an implicit embedding into the dual of the space of Lipschitz functions, we represent the metric differential in terms of partial weak* derivatives. As an application, we show that every mapping in the metric Sobolev space $W^{1,p}(\Omega;X)$ is metrically differentiable with respect to the topology induced by the Sobolev norm, and that in the case $p>n$ it is metrically differentiable almost everywhere.
