a simple response from a simple pilot...
When one derates an engine, the 'datum' from which the thrust calculated from is shifted. For example, B737 engines can be rated at 20,000 lb of thrust, but can be derated down to 18,500 lb. One of the advantages of this is reduced maintenance costs. From this one can reduce the takeoff power setting as far possible given the conditions on the day, in the case of the 737 this is done by fooling it into thinking the ambient air temperature is higher than it in fact is, Boeing calls this the "assumed temperature" technique. If we tell it that the temperature is 56 deg C, the aircraft thinks the air is less dense than it really is, and will reduce the amount of fuel injected into the engine, therefore the thrust output will be consequently lower, reducing noise and strain on the engine.
With a flat rated engine, it will produce it's rated thrust up to it's rated temperature. It won't produce more than it's rated thrust below that temperature - the thrust output against ambient air temperature curve is flat up to the engine's rated temperature, thereafter output is reduced due to falling air density.