At least on uniform grids, finite difference operators become Toeplitz matrices.

Convolutions, too, are Toeplitz matrices on finite sequences.

Now, the fact is that Toeplitz operators, (i.e. infinite dimensional Toeplitz matrices) commute, as they share the same (Fourier-type) eigenvectors. This means that for infinite-dimensional systems, you can apply convolutions and finite difference operators in arbitrary order.

For finite dimensions this is not true, but the "commutativity" still holds at interior points. Thus it's only on the boundary that non-commutatiivity shows up. You will have to create a small boundary correction, but if you have very long data sequences, this is not such a big problem, as convolutions and finite difference operators are "almost commutative."

So the answer is that you may apply the operators in any way you prefer, if you only pay attention to what happens at the boundaries.

Dear Gustaf Söderlind ;

Thank you so much for your relevant replay;

I can't improve on the answer above.