In a multi engine type it is imperative you immediately feather the failed engine prop especially if the failure is a severe mechanical one allowing the prop to quickly reduce below 800 rpm when in some types the prop will not now feather. If you have take off flap set and you retract it close to the deck as part of your take off engine failure drill you may have a significant nose down trim change and fly back into the deck. So in that case don't go blindly into dead foot dead side mantra, flap up gear up automatically as it all depends on the aircraft POH.
Every second you delay feathering the prop could be absolutely vital to a successful climb out. Don't always blindly believe what your instructor tells you - remember he got his info from his old instructor who could be wrong. Do your own research and be guided by the POH. Slow pulling back of the suspected dead throttle can also lose you precious feathering time so you have to get it right first time.