Do the Large Hadron Collider take earth's rotational speed?

[Nugatory]

What is this formula called? How can I derive it?

It's called "Relativistic velocity addition", and a search for that phrase will find some derivations online.

 

[DrGreg]

What is this formula called? How can I derive it?

Wikipedia calls it the velocity-addition formula, but a better name would be "velocity composition formula", as it's not really "addition".

 

[llynne]

May I ask, if two bodies travel at the speed of light, in opposite directions and we calculate the increase of distance between them over time, would they be moving apart at twice the speed of light or is this mistaken?

 

[mfb]

Bodies cannot travel at the speed of light.
If they travel close to the speed of light relative to us and we calculate the increase in distance for our reference frame, we get nearly twice the speed of light, indeed. This is not a relative velocity between two objects, however. If those objects calculate their increase in distance, they get a value below the speed of light.