Billy Davies continues to learn lessons in the harsh and unforgiving environment of the Premier League as Derby’s latest defeat was sealed by a forward whose transfer fee exceeded that of his spending for the summer.

Davies spent about £10 million after Derby’s promotion to the top flight, but that is £1 million less than David Moyes, the Everton manager, paid Middlesbrough for Yakubu Ayegbeni.

The Nigeria striker has since struggled, but his ability to dispatch his only real chance of a moribund match, adding to Mikel Arteta’s opener, will again provide proof for Davies of the increasing gulf between this division and the Coca-Cola Championship.

It is nearly six hours since Derby scored and the problem will not be solved until the January transfer window. By that stage it may be too late.

“We were beaten by a better side and there was no spark,” Davies said. “We’ve got a long way to go until we can compete with teams at this level. We can only work with what we’ve got until January, but I always thought the second transfer window would be very important.”

Everton’s league form has been a concern for Moyes, so a trip to Pride Park was timely. Derby’s football early on was tidy without being threatening and Giles Barnes, the teenager making his first start since April after knee and foot injuries, demonstrated why West Ham United pursued him so vigorously in the summer. But the lack of a goalscorer is glaringly apparent and Davies will be ready to address the problem in the new year, with funds expected to be available.

It was Yakubu’s clever dummy, from Phil Jagielka’s pass, that allowed Arteta to glide through a static defence and dispatch a low shot under Stephen Bywater. Everton should have gone comfortably in front, with Joseph Yobo nodding Tim Cahill’s corner wide early in the second period.

With his team possessing the cutting edge of a plastic spatula, Davies introduced Robert Earnshaw, his record signing, and Steven Howard either side of the hour mark. Earnshaw was branded Derby’s fox in the box after his £3.5 million move from Norwich City but has barely figured, while Howard, so influential last season, has struggled to adapt to the higher level.

But the forward who made the impact was Yakubu. Claude Davis failed to cut out the impressive Leon Osman’s pass and the £11.25 million signing beat Bywater to register his first league goal since early September. “When Yakubu gets through on goal, in the main he doesn’t miss,” Moyes said. “We all know what he was capable of at Middlesbrough and Portsmouth and he’s actually a more intelligent footballer than I first thought. He gets goals in patches, so I’m looking forward to this patch.”

Derby County (4-4-2): S Bywater 6 – T Mears 6, J McEveley 6, C Davis 7, D Moore 6 – M Oakley 6, E Lewis 4 (sub: R Earnshaw, 56 5), S Pearson 7, C Fagan 5 (sub: S Howard, 62 5) – G Barnes 6, K Miller 6 (sub: D Jones, 79). Substitutes not used: L Price, D Leacock. Booked: Moore. Next: Aston Villa (a).

Everton (4-4-2): T Howard 6 – P Jagielka 6, L Baines 6 (sub: A Stubbs, 46 7), J Lescott 7, J Yobo 7 – L Carsley 6, L Osman 7, M Arteta 7, S Pienaar 6 – T Cahill 6 (sub: J McFadden, 86), Yakubu Ayegbeni 7 (sub: V Anichebe, 81). Substitutes not used: S Wessels, T Gravesen. Booked: Jagielka. Next: Birmingham City (h).