C'mon - Liverpool do not have the same sort of motivation to beat these teams!
There might be some truth to that, Spurs, Arsenal and City also have a pretty poor record at Anfield so they may up their game against certain teams. We've had enough local lad's in recent years to explain the importance of these games, Barkley, Baines, Anichebe, Hibbert, plus Kenny and Rooney for a short period and it hasn't had any affect on performances.
However it's worth pointing out that Gerrard viewed United as their biggest rivals. Carragher's only contribution to Derbies was committing a borderline sexual assault on Joleon Lescott and he's a closet Blue anyway, he admitted to paying to watch Everton towards the end of his career in the Main Stand and when he retired he spent more time at GP than Castle Greyskull befoe getting his job with Sky. Even Lallana was raised an Evertonian, Owen the same, Mcmanaman is still a Blue and Fowler was also a Blue. So I don't buy the motivation argument.
We've been able to give City, United, Chelsea and Arsenal hard games when we've been pretty poor, why haven't we done to same against them? We nearly beat Chelsea at Stamford Bridge, after playing 120 minutes plus penalities in the Semi Final against United in 2009. We've beat United 4-0 and 3-0 at home, we've beaten City 4-0 and beat them numerous times at home, even away in the past. We've also beat Chelsea and Arsenal quite a few times in recent years. We always seem to reserve our worst performances every season for them. The 2016/2017 season was a classic example, we put in a great performance and beat Arsenal, then lose to them the next week and not really threaten them. Losing 3-1 to them at Anfield, then play excellently against United and nearly beat them.