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I'm studying Weinberg's 'Gravitation and Cosmology'. In particular, in the 'Curvature' chapter it says that the Riemann tensor cannot depend on ##g_{\mu\nu}## and its first derivatives only since:

... at any point we can find a coordinate system in which the first derivatives of the metric tensor vanish, so in this coordinate system the desired tensor must be equal to one of those that can be constructed out of the metric tensoralone, ..., and since this is an equality between tensors it must be true in all coordinate systems.

What I don't understand is how introducing the second derivatives should change this situation. The point is that (and I'm not sure about that...) we can always find a locally inertial frame. In this frame ##g_{\mu\nu}=\eta_{\mu\nu}##, which is constant, and hence all its derivatives should vanish.

What am I doing wrong?

Thanks a lot