Liam Rosenior's departure from Chelsea on Wednesday marked the 10th managerial casualty in England's top flight this season.
Carrick, who took over at United in January following the sacking of Ruben Amorim, said there was a balance to be struck between short-term success and building for the future.
"There are two sides to it," the 44-year-old told reporters on Thursday.
"There are instant results and the next game being important, but there's definitely a responsibility, our thinking of what the future looks like and the bigger picture.
"There are all sorts of what-ifs in this world. Half full, half empty? I like to live my life in a positive way. I don't think of what could go wrong, that doesn't come into it. It's what can be achieved. What success looks like."
United have impressed under Carrick, winning eight and drawing two of their 12 matches to sit third in the league. Six points from their remaining five games would secure Champions League qualification after a two-year absence.