In our airline we use assumed temperature derate and unbalanced field V speeds for each take-off, calculated for each runway (and each runway intersection) using individual charts prepared by our performance ops department.
bizjet pilot, pholosphies about whether or not to increase power on the operating engine folowing an engine failure vary widely. On the BAe 146 fleet my airline's philosophy was to increse thrust to rated folowing a failure, but on the 737 fleet we keep the operating engine at the derated thrust.
The thrust derate process allows for all performance after an engine failure, including the single engine climb, so it isn't necessary to increase the thrust (with the increased flying complications involved in the increased yaw). If you do increase the thrust, though you get a performance buffer - six of one half dozen of the other perhaps?