Mr Derbyshire pleaded guilty to charges of entering the playing area at a designated football match without lawful excuse and common assault against Eddie Howe.
The incident occurred at Elland Road on May 13 this year.
Chief Superintendent Richard Close led the police operation at the Yorkshire ground during that match.
He said: “Criminal behaviour of this kind is completely unacceptable, particularly where club staff or players are targeted during a large-scale sports event such as this.
“I know that fans of the sport, regardless of their club affiliations, will join us in condemning anyone who acts in this way.
“We will always treat incidents like this very seriously and we work closely with Leeds United and visiting clubs to fully investigate any offences and take robust action against those responsible, including seeking football banning orders against them.”
Mr Derbyshire left his seat in the 93rd minute before confronting Howe in the away dugout, before shoving his left shoulder, a court heard.
Security quickly managed to escort the fan away from the scene while the incident was broadcast on live TV.
Leeds have also handed him a lifetime ban from Elland Road.
Speaking to talkSPORT post match, Howe said in May: “For me, I’ve never experienced that before.
I was looking in the opposite direction and he ran on and shouted something at me that I won’t repeat.
“It was over in a flash but it was the last thing you expect in that moment.
“It’s alarming for football really because we can’t have that, the safety of everyone, coaching staff, players, is of paramount importance.”