The two sets of players clashed towards the end of January's match at Pride Park, which the Rams won 1-0.
Both clubs accepted the charge and were each fined £35,000 and warned as to their future conduct. A further £10,000 suspended penalty imposed for a similar case between the two clubs earlier in the season was also activated.
Forest's chief executive Mark Arthur said: "We are surprised at the severity of the fine that has been imposed. We will study the FA's written report before deciding whether or not to appeal."