I think the Liebniz notation is confusing you. Therefore:

    [U(G(W))]' = U'(G(W))G'(W)
    [ln(AW)] ' = ln'(AW) [AW]' = (1/(AW)) A = 1/W

If G(W) = AW, then G'(W)=A.

So in that case [U(P)=ln(P)] your inequality does not hold.

The inequality I gave is a criteria, not a theorem. I didn't claim it was always satisfied, I just claimed it was possible.

If it is satisfied (e.g., U(P)=sqrt(P)), a NIT creates an incentive for work. If equality holds (U(P)=ln(P)), a NIT has no marginal effect on work. If the reverse inequality holds (U(P)=1-1/P), a NIT creates a disincentive for work.