ALL available fares are always shown on the National Rail website. Using other websites is normally totally pointless. However the only exception is where 'Megatrain' fares are available on the route. That only applies to certain services operated by Stagecoach but Derby to London is one of them. So, to find the cheapest fares, I've needed to look at both the National Rail website and the Megatrain one. (As I've said, looking anywhere else is totally pointless). Here are the results of my search:
There are no early trains with cheap fares available.
The earliest you can get a sub-£40 return fare (and only by a pound) is at 1001, arriving at 1133. (That's £27 single). You could then return at 2130 (arrives 2358), 2225 (arrives 0041) or 2315 (arrives 0218), for £12 single, making £39 return in total. (NB: See below for a way of cutting this return fare, by combining both websites).
To get a cheaper fare you'd need to leave at 1101, arriving at 1233, for £21. Returning as above, your total return fare would be £33. (NB: See below for a way of cutting this return fare, by combining both websites).
All of the above can be booked via the National Rail website.
http://www.nationalrail.co.uk/
Using the Megatrain website there aren't any available outbound trains until 1201 (arriving 1334). That's just £15 per person single, so you could combine it with the return services above for a total return fare of £27 per person.
However that final train from London (at 2315) is available for just £5 single per person on the Megatrain website, so (if you didn't mind the late arrival into London and the really late return into Derby) you could get the return fares down to £20 each. Alternatively you could book the outbound journey, at 1001 or 1101, with the National Rail website, and the return journey at 2315 with Megatrain, giving return fares of £32 (arriving in London at 1134) or £26 (arriving in London at 1234).
Chris