(See attached file for full problem description)
I assume that what you claim is TRUE and all you need is a few pointers rather than full solutions.
Here are the pointers:
1a+b: Write a vector product via the antisymmetric unit tensor e_ijk:
(axb)_i = e_ijk*a_j*b_k
and either open the brakets for (a) or differentiate for (b).
If you have not studied about e_ijk (sometimes it is epsilon_ijk rather than e_ijk),
write the vector product as a determinant with the 1st row being the unit vectors i, j, k, the second row being a_x, a_y, a_z, and the 3rd row being b_x, b_y, b_z. Then handle the 3rd row as a sum of two rows for (a) or differentiate for (b). In the second (determinant) version you will have to figure how to differentiate a determinant, so I would recommend the e_ijk approach as the easiest if ...