  We represent minimal upper gradients of Newtonian functions, in the range
$1\le p<\infty$, by maximal directional derivatives along "generic" curves
passing through a given point, using plan-modulus duality and disintegration
techniques. As an application we introduce the notion of $p$-weak charts and
prove that every Newtonian function admits a differential with respect to such
charts, yielding a linear approximation along $p$-almost every curve. The
differential can be computed curvewise, is linear, and satisfies the usual
Leibniz and chain rules.
  The arising $p$-weak differentiable structure exists for spaces with finite
Hausdorff dimension and agrees with Cheeger's structure in the presence of a
Poincar\'e inequality. It is moreover compatible with, and gives a geometric
interpretation of, Gigli's abstract differentiable structure, whenever it
exists. The $p$-weak charts give rise to a finite dimensional $p$-weak
cotangent bundle and pointwise norm, which recovers the minimal upper gradient
of Newtonian functions and can be computed by a maximization process over
generic curves. As a result we obtain new proofs of reflexivity and density of
Lipschitz functions in Newtonian spaces, as well as a characterization of
infinitesimal Hilbertianity in terms of the pointwise norm.
