Even though they had waited three years to beat Manchester City at Old Trafford and far longer to do so with such authority, Manchester United's emphatic victory in last weekend's derby was the easy part. Any team, especially one with a new manager who is steeped in the club's glorious history, can get up for a match with their biggest rivals and win it.The real challenge for Michael Carrick and United is to back up their dominant performance against the second-best team in the Premier League when they face the very best, Arsenal, at the Emirates Stadium on Sunday. And it is even more important that they stay motivated when it comes to their next home fixture against Fulham and then the midweek trip to West Ham that follows.Carrick might have gotten off to a dream start, but the journey to the all-important goal of qualifying for the Champions League has just begun, and it is going to be an arduous one. Arch-rivals Liverpool are one point and one place above United in that all-important fourth spot that guarantees a place in Europe's top competition. Finishing fifth might also be enough to seal Champions League football given the dominance of English teams in continental competitions, but United cannot afford to take any chances and rely on results elsewhere. They have been away from the Champions League for too long and need to be in charge of their own destiny.United's failure to take advantage of other side's failings and enter the top four led to Roy Keane and Jamie Carragher writing off their Champions League hopes as recently as last month. But that was when Ruben Amorim was in charge, and Carrick is now riding a wave of optimism. Keane was telling his Sky Sports colleagues to "calm down" after the derby win, but the stars are aligning for United to build some momentum and seize that top-four spot.GOAL lays out the six reasons why United should, and will, finish in the top four after all...While Carrick is definitely less charismatic than his predecessor Amorim, he has shown to be much bolder tactically. And if United continue to embrace their adventurous side then they can accelerate towards their objective by virtue of winning more games. Some Amorim defenders pointed to the fact that United had lost only one of their previous eight games when he was sacked, but the team were dropping too many points by drawing matches they should have won. The coach tended to play it safe, rarely making in-game tactical tweaks even when it was glaringly obvious he needed to, such as against 10-man Everton as well in the draws with Wolves and West Ham at Old Trafford.What galvanised United fans so much on Saturday was how the team didn't want to simply beat City, but they wanted to destroy them. Going for the jugular is key to United rolling the weaker teams over, and even if they lose the odd game they might have drawn with Amorim's approach, it should lead to them earning considerable more points than playing with the handbrake on.While Amorim thought he was playing the long game and would be shown patience, Carrick knows he is on borrowed time already as the club are already on the lookout for a permanent coach. His only hope of landing the job full-time is to get United back into the Champions League, and even then it might not be enough. It means he has nothing to lose, which should make his team more thrilling and more threatening.Carrick did not pull off a tactical masterclass as such against City, with the simplest and most obvious of changes being enough to propel United to victory. The most effective decision was to play Bruno Fernandes in attacking midfield rather than in the more withdrawn role that he played under Amorim. Darren Fletcher had also given his captain freedom to get forward in his two games as interim boss, and on Saturday - for the third game - running Fernandes picked up an assist. He was at his very best against City, creating four big chances in the game, and while he did pull off the same feat under Amorim, that was against lowly Wolves, rather than against one of Europe's best teams. No player had created as many chances against Guardiola's side for a decade.Fernandes was not the only player to thrive in his natural position. Amad Diallo was restored to right wing rather than being forced to play wing-back, meaning he didn't have to defend one-vs-one against Jeremy Doku as he previously would have. Harry Maguire and Lisandro Martinez also benefitted from returning to a back four, where they struck up a formidable partnership which ensured Erling Haaland had no shots.Kobbie Mainoo was another who excelled in a more familiar role than under Amorim, playing in the 4-2-3-1 shape he had shone in when breaking into the first team under Erik ten Hag. In fact, just seeing Mainoo out on the pitch, let alone the starting line-up, was an unfamiliar but hugely welcome sight given Amorim's apparent disdain for the England international.Mainoo felt important again after being pushed to the brink of leaving his boyhood club by the Portuguese, and it showed as he delivered a high-octane performance, playing a key role in United's aggressive pressing plan as well as adding some flair in possession. Mainoo made 77 high pressures in the game, according to Opta, the fourth-highest of any United player all season, and he now possesses the second-most pressures per 90 minutes in the Premier League, only behind Everton's Merlin Rohl.Mainoo's performance against City showed how wrong Amorim had been to overlook him for so long, but one great performance does not make a great midfielder and he has to keep on producing. Fortunately, he has a point to prove after so long in the wilderness and will be keen to back it up.Add GOAL.com as a preferred source on Google to see more of our reportingAmorim might well argue that he lacked fortune in his final few games in charge, with Fernandes succumbing to a very rare period of absence while Mainoo and Mason Mount also picked up injuries to join longer-term absentees Maguire and Matthijs de Ligt on the sidelines. Factor in this all happened while Bryan Mbeumo, Amad and Noussair Mazraoui were at the Africa Cup of Nations, too, and United's squad was severely depleted around the turn of the year.Then again, Amoriom had close to a full squad for the first four months of the campaign and couldn't make the most of it.Still, you could say that Carrick was fortunate to be able to count on Mbeumo, Amad, Maguire, Martinez, Mount and Mainoo in his first game, and their continued presences will only increase the belief that he can take United into the top four and keep them there. Mazraoui missed the derby as he was preparing for the AFCON final with Morocco but he will be back to face Arsenal, leaving only De Ligt and Joshua Zirkzee in the treatment room. The Dutch pair are due back in February, and if no players pick up fresh injuries before then, then Carrick will be armed with a fully-fit squad for the final four months of the season.While injuries can happen at any moment, Carrick has a better chance of his squad staying fit than the other coaches he is vying with for that highly-coveted top-four berth as United only have the Premier League games to concentrate on. He has Amorim to thank for that as the Portuguese failed to get United into Europe - for the first time in 11 years - and oversaw their eliminations from the Carabao Cup and FA Cup at the first hurdle, something that had not happened since the 1981-82 season.United will play their fewest amount of matches in a season since before the First World War (40), and that light schedule gives Carrick a massive advantage over every other team with an eye on qualifying for the Champions League. Arsenal, City, Liverpool, Chelsea and Newcastle are all still involved in the FA Cup and Champions League, with the Gunners and Cityzens also set to play at least one, and almost certainly two more games in the Carabao Cup.Aston Villa, meanwhile, have Europa League and FA Cup commitments, leaving Everton as the only team still with a realistic chance of rivalling United for a top-four place who only have one objective to worry about.It means United's squad should be fresher than the rest while they will also have more time to prepare game plans on the training ground.This season has witnessed the most competitive race for Champions League football in Premier League history, with just six points separating Liverpool in fourth spot and Brighton all the way down in 12th. It means no team can be taken for granted, as United know all too well after failing to beat all of the teams in the bottom six at least once. Yet it also means that every side around them have their problems. While Arsenal, City and Aston Villa should be assured of finishing in the top three and stamping their ticket back into the Champions League, fourth and fifth spot are well and truly up for grabs.Liverpool have failed to win in their last four league matches meaning that Arne Slot is not far from being sacked while Chelsea, who are one point and one place below United, have a near novice coach in charge in Liam Rosenior and a talented but mismatched squad that is severely lacking experience.Seventh-placed Brentford are the surprise package of the season, and even being in the conversation is an incredible achievement for the Bees given it is Keith Andrews' first season as a coach and he lost so many key players in the summer, including his two top scorers and captain. But Brentford have lost 40 percent of their games and have little experience of fighting for European spotsNewcastle, level on points with the Bees in eighth, are struggling for consistency and worn out from juggling Champions League football while Sunderland, also on 33 points, lack the true quality to sustain a top-four charge. As for Everton, Fulham and Brighton below them, they all seem to have too many flaws to put together a consistent run of victories.