By half time, Hull found themselves facing a 4-0 deficit with Sebastian Larsson also missing a penalty, before the Rams completed the rout after the break.
The Tigers now sit in 10th place in the Championship ahead of the remainder of the weekend's fixtures, and Slutsky has admitted that his players need to provide a positive response in their next game.
The 46-year-old is quoted by BBC Sport as saying: "First of all, I must apologise to our supporters because they don't deserve this result. I'm sorry because I prepared the team, it's my fault when the team lose.
"The next match will show how we react to unbelievable stress. For me, it's most important to concentrate on the next match and prepare very hard on the next match. Now, for me and for the team, I think it's a key moment in this season."
Hull are next in action when they play Fulham at Craven Cottage on Wednesday, September 13.