Since the Earl of England, Edward, coined the word "derby" more than 200 years ago, the word, which originated in horse racing, has been associated with rivalry, passion, speed and competition, and when invoked on the green field, it has become synonymous with the "Battle of the Cities! It is synonymous with the "battle of the cities".
L.A. here is short for Los Angeles, which has two NBA teams, the Los Angeles Lakers and the Los Angeles Clippers.
The L.A. Derby is when these two teams meet in an NBA game!