The Spaniard conceded two questionable goals on his competitive debut for United in Sunday's Community Shield win over Manchester City.
De Gea failed to react quickly enough to Edin Dzeko's 30-yard strike and was arguably at fault for not coming to collect the cross that led to City's opener in the Wembley clash.
His blushes were spared by a second-half rally which saw the Red Devils run out 3-2 winners, but Sir Alex Ferguson's ?18million gamble will be questioned if such errors are repeated throughout the season.
When asked whether he was experiencing problems adapting, the 20-year-old said: "No, football is football here and in any country.
"I have to adapt quickly and improve on the whole of my game."
De Gea, who signed a five-year contract at Old Trafford following his move from Atletico Madrid in June, brushed off the importance of conceding two first-half goals.
He said: "The two goals we conceded in the first half stopped us in our tracks but in the second half we were able to claw back and I think that's the important thing. I'm very happy about the victory."
Chris Smalling tapped home United's first before a Nani brace allowed the Premier League champions to take the spoils in a hotly-contested derby at Wembley.
The highlight of the game was the Portuguese star's first goal - a wonderful tight-passing move which ended with the winger neatly clipping the ball past City stopper Joe Hart.
De Gea was impressed with the calibre of the goal, adding: "It was a beautiful goal and the lads in the dressing room were very happy."
Source: Team Talk
Source: Team Talk