a question about exercise 21.7

Although it is easy to see that the answer comes from the fact that d/dt = multiplication by H, which has no x dependence when H is spatially invariant, I seem to find a contradiction: since mixed partials commute, we always have d/dt d/dx = d/dx d/dt, but p = ih_bar d/dx, so doesn't that imply p always commute with d/dt? Thanks in advance!