City went down to a goal from ex-target Chris Porter in an A50 derby.
After last Thursdays goalless draw with Notts County, and Saturdays 6-0 win over Newcastle Town, this represented the major players within the squads third game in five days and it possibly showed early on as Derby took control.
City named just 16 men in their squad, hoping to give the players more game time, but found themselves behind after eight minutes when Roberts cross was headed in from close range by Porter.
It took the Potters a while to get into their rhythm but when they did they threatened the Rams goal, Mama Sidibe heading wide from a couple of yards out after meeting a teasing cross from Delap.
Sidibe went close again before half time but headed wide again, this time the assist coming from Danny Collins.
The Potters lost Matthew Etherington just after half time, to a groin niggle, but the injury is not thought to be serious. Thomas Sorensen, Liam Lawrence and Tuncay were amongst players to come off the bench for the Potters.
City continued to look the more dangerous and Delap ballooned his effort over the bar, after collecting a flick on from Sidibe on the hour mark.
Cywka, a summer signing from Wigan, nearly doubled the Rams advantage but his slide was inches away from converting Ben Pringle's low cross-shot.
Commons lashed a shot across the box shortly after before the best City chance of the game, as Tuncay turned inside the box, got himself half a yard, and watched as Bywater instantly dropped to his right to push the Turkish stars shot away one handedly.