Hi Andy. To get the idea of de-rate lets forget about FMC and clever software.
First lets think about a modern Gas Turbine.
It has a normal TO thrust.
This thrust lets say is 20000lb.
So the engine is designed such that when the pilot or autothrottle system puts the thrust levers to the TO position the engine accelerates to a speed that deliverers 20000lb.
However if the air temperature is too high(density too low) the engine will reach its maximum operating temperature (TGT, ITT, EGT) before it reaches the speed that equates to 20000lb thrust. The engine is now temperature limited and the maximum thrust it can produce reduces as the air temperature increases(density reduces).
lets say that at all temperatures below and up to ISA (+15 at sea level) our engine can produce 20000lbs but as the temperature becomes higher than ISA the thrust reduces, Say 19000lb at isa +5 (+20) 18000lb at ISA +10 (25).
Today the temperature is below ISA so I will get 20000lb thrust when I set Take Off thrust.
My runway is 10000ft long.
So I calculate that today I can take off with a weight of 35000kg and achieve all the take off requirements.
I find that my actual take off weight today is in fact 30000kg.
I have more thrust than I need to achieve all the requirements. So I can choose to use all 20000lb of thrust and have very very good performance or I can use less than normal TO thrust and just meet all the requirements.
I choose to reduce the thrust to save engine life (money)
How much thrust do I need?
I calculate that at my weight I only need 19000lbs of thrust.
How do I set 19000lbs of thrust? If I put the thrust levers to TO because the temperature is low I will get 20000lb. I could put the thrust levers somewhere below the TO position but the problem with this is it's tricky to set them exactly and it will take time to adjust them to the correct setting.
But I know that if the temperature was isa +5 (20) if the thrust levers are at TO I would get 19000lbs.
So I cheat and tell the engines its ISA +5. So I can put the thrust levers to the TO position and I will get the 19000lb thrust I need.
This way I always use the TO position for take off. And I can use this method to always set just the amount of thrust I need to take off no matter what my weight is.
Notes.
1) The engine has to do a bit of clever calculation because even though I told it the temp was 20, the air entering the engine is 15.
2) There are maximum limits to how much I can reduce thrust so some time I will have a lot more thrust than I need.