Every silver lining has a cloud and the good-news bulletins that followed the arrival of four English teams in the Champions League quarter-finals were swiftly followed by this terrifying reminder that the gap between top and bottom in the Barclays Premier League is seismic, fissure-like and growing.

Chelsea scored six, Frank Lampard alone got four and the truth is that either tally could have doubled, such was the superiority of Avram Grant’s side.

The difference was embarrassing. As Lampard carved out a second-half hat-trick in 15 minutes that also included a goal from Joe Cole, anyone who cares for football would have felt uncomfortable with this preposterous walkover. Lampard is an exceptional player, but, even so, this is not right. This is not how it is meant to be.

So much of the game had the look of a training-ground exercise — and some of the goals, too. Previously, looking at Derby County’s forlorn league record, one could be forgiven for asking if they really are as bad as that. Sadly, the answer is yes. Paul Jewell inherited a team well on their way to being confirmed as the worst in Premier League history and has not been able to improve them.

Deep down, Chelsea will know the ultimate meaninglessness of this scoreline, too. It is not even Derby’s worst defeat of the season, having lost by six without reply to Liverpool.

Certainly, Chelsea’s fans understood. The win kept their title hopes alive, but they still left early to avoid the traffic. There was no hint of the jubilation that would traditionally greet such a result. What did you expect? It was only Derby.

Lampard looked happy, though, and well he should. Even the frailty of the opposition should not detract from his feat in scoring three goals from midfield, plus a smartly taken penalty, which he also won. Indeed, the strangest thing about Chelsea’s win was that only one of the six goals was scored by a striker, Salomon Kalou, and even then he was playing in a wide attacking role. Nicolas Anelka failed to get on the scoresheet; so, too, Didier Drogba and Andriy Shevchenko, the substitutes.

The rout began in earnest after 26 minutes when Dean Leacock, an early replacement for the injured Alan Stubbs, whose experience was sorely missed, tripped Lampard in the area. The England midfield player stepped up and struck his shot to Roy Carroll’s left. Derby heads went down and did not lift, much to Jewell’s fury.

efore half-time, Chelsea got a second goal and the cushion that enabled them to play with such ease after the interval. Lampard was the architect, slipping a pass through to Anelka, who was thwarted by a clearance from Carroll. The ball travelled only as far as Kalou, who hit it, first time, from 25 yards into an unguarded net.

“The fourth official has indicated one minute of added time,” a stadium announcer said as the half-time whistle loomed. To be fair, he could have indicated decent Thai restaurants in the Fulham area or recommended a good book; the contest all had paid to see was long over.

The second half belonged to Lampard. In the 57th minute, he converted a tap-in from a cross by Joe Cole as Derby ball-watched and nine minutes later he struck a shot from the edge of the area that Carroll could not stop. His fourth, in the 72nd minute, was another medium-range effort, perhaps the best of the night, cutting inside and riding two challenges after a pass from Joe Cole. They will miss him more than they know if his contract talks again break down. Between Lampard’s second and third, Joe Cole also scored, first to the loose ball after Claude Makelele had set Anelka clear and Carroll had saved.

“Frank Lampard is an important player for the team. What more can I say, he always scores,” Grant, the Chelsea first-team coach, said. “I am sure he will be here next season. If Frank says he wants to stay at Chelsea and Chelsea say they want Frank to stay, what is the problem?”

What indeed? David Jones, a substitute, did at least put Derby on the scoresheet, but the event was treated more with amusement even by the home supporters, another of many worrying signs on a night that was good for Chelsea but disquieting for the rest of us.

“They looked like championship heavyweights, we looked like ABA lightweights,” Jewell, the Derby manager, said. “Everything they needed to do they did better than us. I was squirming again for 90 minutes. It comes to something when you are happy to settle for six. I’ll make allowances for ability, but I find it difficult to accept that we cannot match their desire.”

It gets no easier. Derby’s next match is against Manchester United on Saturday. “I’ll see you there,” a journalist told Jewell, cheerily. “I might not turn up,” the manager replied, no doubt wishing he had that option.

Chelsea (4-3-3): C Cudicini – P Ferreira, R Carvalho, J Terry, A Cole – F Lampard, C Makelele, M Ballack (sub: D Drogba, 67min) – J Cole (sub: A Shevchenko, 74), N Anelka, S Kalou. Substitutes not used: Hilário, M Essien, Alex.

Derby County (4-5-1): R Carroll – M Edworthy, D Moore, A Stubbs (sub: D Leacock, 15), J McEveley – M Sterjovski, H Ghaly (sub: R Earnshaw, 52), S Pearson, R Savage, E Lewis (sub: D Jones, 58) – K Miller. Substitutes not used: L Price, E Villa. Booked: McEveley.

Referee: C Foy.