Jürgen Klopp looked around a distraught Liverpool dressing room at Wembley in February and urged his team to use defeat in the League Cup final as motivation to go one better in future. The message continues to resonate on the evidence of a comfortable and comprehensive defeat of Derby County.

Encouragement was not confined to the manner or margin of victory for the Liverpool manager. Ragnar Klavan scored his first goal for the club, Emre Can returned from injury, Danny Ings made his first appearance of the season, Marko Grujic and Loris Karius were handed competitive debuts and promising academy youngster Ovie Ejaria graduated to the senior ranks as a late substitute. But it was the class of Philippe Coutinho and, increasingly, Roberto Firmino that made the deepest impression. The Brazilian duo dissected the Championship team in a commanding second half display illuminated by fine goals from Coutinho and Divock Origi.

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“Could we have done better? Yes. Is that important? No,” said a satisfied Liverpool manager. “You need to be 100% professional and serious in games like this, which can be difficult. The opponent was ready for each opportunity but we didn’t give them any. In the first half we could have been better with the last pass and in the second we could have been better with the counter attack but we scored the goals. This is a very important tournament for us, not only because I say it but for the players too and we have to show it is important from the first second. It is important so let’s go for it.”

Liverpool have lost only one of their last 14 League Cup ties against lower league opposition and, although their threat and authority was immediately apparent at the iPro Stadium, Nigel Pearson was right to be aggrieved at how the visitors took the lead.

Derby had settled into the contest when a serial lapse in concentration showed their problems are not confined to scoring goals this season. Four home players were found wanting as Coutinho’s corner whipped in from the left. Two missed the flight of the ball at the near post, Richard Keogh sliced what should have been a routine clearance behind them and Klavan, Liverpool’s defensive recruit from Augsburg, reacted quicker than Cyrus Christie to convert via a slight deflection off the full-back.

With Firmino tormenting Keogh and Christie down the Liverpool left and Derby’s defence struggling to rediscover their poise, the visitors could have extended their advantage before the interval with greater composure on the break. Origi forced a good save from Jonathan Mitchell, the young goalkeeper who impressed on only his second appearance for Derby in place of the ill Scott Carson, while Coutinho was also denied from a dangerous free-kick.

Derby’s prospects of an equaliser always seemed remote with Darren Bent isolated against Klavan and Joel Matip. They were extinguished early in the second half when Coutinho and Firmino turned on the style and gave the scoreline a more accurate reflection of the chasm between the teams.

Liverpool’s number ten started and finished the move that doubled the visitors’ advantage. Collecting the ball in central midfield, Coutinho spun away from his marker, found Firmino and continued his run through the heart of the Derby defence as his compatriot advanced on goal. With barely a glance to his right, Firmino rolled a perfectly-weighted pass into the area where Coutinho arrived on cue to side-foot beyond Mitchell from 12 yards.

Klopp’s team, and in particular their Brazilian forwards, toyed with the lowest scoring team in English league football and added a third shortly afterwards. Firmino was involved in the build-up but it was Coutinho who this time delivered the defence-piercing pass, flicking the ball into space on the right for Origi to finish emphatically into the roof of the net. Coutinho created another clear opening for Firmino but Mitchell denied the former Hoffenheim forward at close range before Derby’s tormentor-in-chief was withdrawn to a standing ovation from the travelling section.

“We were beaten by a very good side,” admitted the Derby manager. “The gulf between the two sides at the moment is too big for us. At 3-0 down after an hour it can be very difficult but we continued to work extremely hard. There is a big gulf between where we aspire to be and where we are at present. When you are on the end of a defeat like this there has to be some honesty about it. We were well beaten by the better team and we made mistakes that cost us quite dear.”