I recently read an article about derated takeoffs and I understand the concept, but how do pilots know what temperature to put into their FMS if they want an FLX takeoff? Some people say it's a rough estimate like ~50-60 C, but is this true? If it is gauged by some external factor, what is it?
For Airbus Aircraft you use a tool from Airbus where you put in data like wind, air temperature, pressure, weight, runway characteristics (length, heading, elevation, etc) and other elements that calculates the flex temperature. It also gives you some other parameter like required runway length among others.
The principle is that you run the take-off distance calculation in reverse.
There are tables or graphs in the QRH that tell you what runway distance you need for take-off given the pressure, altitude, temperature, head-wind, cross-wind and runway state (dry, wet, snowed…). And you put in all the other numbers and then look up the highest temperature that make you still fit in the runway distance actually available, usually with some margin. Or the highest temperature allowed—you don't derate below the normal climb thrust.
These days most pilots will have an "electronic flight bag", an application on notebook, tablet or in newest aircraft even built-in computer in cockpit, where they'll put the numbers and it will look up the values for them. But the principle is still the same: find the highest temperature for which the available runway is sufficient.