The reigning champions have recorded five wins and a draw from their opening six matches to sit one point clear of Old Firm rivals Celtic but Davis believes that the output of the 'Gers players will only get better.
"The main priority today was to leave with three points going in to the Old Firm game and we knew that Dundee United would be well organized," he told the club's official website after Saturday's 1-0 win over the Terrors at Tannadice.
"Even when they went down to ten men in some ways it was a bit harder to break them down but we got the goal and probably should have added to that.
"I probably should have scored but I thought I had done everything right. It came off both posts and they had a little bit of luck with it."
As for dropping out of the Europa League prior to the group stage, Davis insists it has been a positive in the start to the club's SPL title defence.
"The European side of things was a big disappointment but we have been fully focused domestically and it has been quite a difficult start to the season in terms of away fixtures," he added.
"We've always managed to pick up results even when we haven't played at our best, like the Aberdeen game a few weeks ago.
"I still think there is a lot more to come from the squad in terms of our general level of play but as long as we keep picking up wins along the way that will come.
"We've managed to come out of that top of the league so we can go in to the Old Firm game with confidence and hopefully pick up another result."