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We represent minimal upper gradients of Newtonian functions, in the range$1\le p<\infty$, by maximal directional derivatives along "generic" curvespassing through a given point, using plan-modulus duality and disintegrationtechniques. As an application we introduce the notion of $p$-weak charts andprove that every Newtonian function admits a differential with respect to suchcharts, yielding a linear approximation along $p$-almost every curve. Thedifferential can be computed curvewise, is linear, and satisfies the usualLeibniz and chain rules.  The arising $p$-weak differentiable structure exists for spaces with finiteHausdorff dimension and agrees with Cheeger's structure in the presence of aPoincar\'e inequality. It is moreover compatible with, and gives a geometricinterpretation of, Gigli's abstract differentiable structure, whenever itexists. The $p$-weak charts give rise to a finite dimensional $p$-weakcotangent bundle and pointwise norm, which recovers the minimal upper gradientof Newtonian functions and can be computed by a maximization process overgeneric curves. As a result we obtain new proofs of reflexivity and density ofLipschitz functions in Newtonian spaces, as well as a characterization ofinfinitesimal Hilbertianity in terms of the pointwise norm.

Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential