Derry to Liverpool your choices are Ryanair, Ryanair or Ryanair I think (unless you wanted to drive to Aldergrove where Easyjet fly from)
Flight times are crappy as well it seems .. only in the middle of the day which wouldnt' really suit for commuting weekly as you'd miss most of Friday and have to go back very early on Sunday
Book well in advance ... taking full advantage of hte periodic 99p etc. sales ... but without any sales i'd say you're likely looking at paying between £50 and £100 a week.