There are very different formulas for the differential cross section that depend on what is more convenient for your particular case. However I found often usefull the formula for the differential cross section of a process "two particle in many particles":
where M is the matrix element of the process.
The total cross section is obviousli the integral of the differential cross section. However you usually define it as:
$$dN_r=dN_f \cdot \sigma \cdot n_b\cdot d$$
where [itex]N_r[/itex] is the number of particles produced by the reaction, [itex]N_f[/itex] is the number of particles in the beam, [itex]n_b[/itex] is the density of targets and d is the thickness of the target.