Derry took charge with the Meadow Lane club bottom of League One and he has now lost his first two games at the helm, conceding eight goals in the process.
County took the lead at Oldham, but their defence was ruthlessly exposed and they also saw captain Dean Leacock sent off.
Derry's only consolation was the 13th-minute goal scored by Ronan Murray, who marked his first County start with a neat finish.
Leacock was sent off after 55 minutes, adjudged to have hauled back Jonson Clarke-Harris just outside the box and shown a straight red card.
Oldham ran riot as Dayton slotted in their third from a Philliskirk pass and then returned the favour to Philliskirk, who slammed in his second of the night.
Latics substitute Jordan Bove was sent off just three minutes into his debut - and on his 18th birthday - for a high tackle on Kieron Freeman, but Oldham completed the rout in stoppage time when Clarke-Harris snapped up a rebound.
Derry said: "I have to apologise to the fans who came up here. There aren't many times in my career that I've been speechless, but this is one of them.
"I'm new to management and this will be my most valuable learning experience because I will make sure it doesn't happen again.
"That performance was not acceptable. The players need to look at themselves in the mirror and ask if they are being the best professionals that they can be.
"We are at a crossroads and I know the direction I'm heading in, so it's up to the players whether they want to come with me.
"In times of adversity, the men start to stand up. I've seen them do it in training and I don't want players going into hiding when they set foot on the pitch."
- Sports & Recreation
- Danny Philliskirk
- Dean Leacock
- Korey Smith
- James Tarkowski