Manchester derby: Robin Van Persie breaches City's fortress
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Van Persie had other ideas, however, and stepped up in stoppage time to fire a free kick past Hart with the aid of a slight deflection off Samir Nasri.
"It's very hard to lose a derby, but even more so in the last minute so I'm very disappointed, said Mancini.
"But we have the players and the spirit to recover and I believe we will still win the Premier League and FA Cup double."
Tottenham were close to a fourth consecutive league win that would have cemented their place in the top four when Clint Dempsey opened the scoring at Goodison Park after 76 minutes.
Everton, beaten only twice in the league this season, equalised with a header from their former Tottenham player Steven Pienaar before Jelavic poked home the winner.
Liverpool had to make do without Luis Suarez, their only genuine striker for the trip to West Ham, but proved they can still prosper without the Uruguayan.
Johnson's piledriver put Liverpool ahead in the 11th minute but West Ham were in front by halftime through a Mark Noble penalty and a Steven Gerrard own goal.
Cole's composed finish levelled matters after 76 minutes before Jonjo Shelvey, another player who learned his trade at West Ham, went through on goal with 10 minutes left and forced James Collins into the game's second own goal.