Hull were four down at half-time and the Russian manager described the result as the worst of his career.
Hull have not won an away League game since the start of last season but they could not have envisaged how bad this defeat would get, even after Matej Vydra drove a 20-yard free-kick past the defensive wall in the 15th minute.
It might have been different if Sebastian Larsson had converted a penalty when Tom Lawrence hauled over Ola Aina in the 27th minute but he blazed high over the bar and the rest of the game turned into a rout.
Vydra made it two when David Nugent set him up in the 34th minute and former Hull captain Curtis Davies headed in on his 400th league appearance before Bradley Johnson tapped in unmarked at the back post on the stroke of half-time.
Johnson was presented with his second by an awful clearance from Michael Hector in the 58th minute leaving Slutsky to say: “I apologise to the supporters because they don’t deserve this result.
“I’m very disappointed and sorry, I prepared the team and it’s my fault when it loses. It’s now a very difficult and important time because next match will show how we react to unbelievable stress.
“For me and for the team it’s a key moment in this season. If I think about the past we won’t have a future so for me it’s history, not good history for me because it’s the biggest defeat in my coaching career and now I am only thinking about the next match.
” I am not a robot, I am human and for me now it’s a very difficult time and I am trying to keep a responsibility in this difficult situation.”
Derby manager Gary Rowett wanted a response to the defeat at Sheffield United before the international break and he said: “Tonight was a really strong reaction.
“You could tell quite early in the game the mentality was right both in possession and out of possession and I think the difference tonight was we finished off the chances we had, which we haven’t done in other games.
“The quality was really good but the challenge for us is can we put three good performances together and for me we’ve got to find that mentality after those three games rather than after a defeat, that’s the key.
“I thought the penalty miss was a big moment, I felt that was something that changed the game a little and gave the team and the crowd an extra lift.”