I've also been referring to the National Rail website.
The 'walk-up' single fare (Standard Open Single) between Derby and Edinburgh is �69.50. The Standard Open Return fare is double that, at �139, but (as long as you can travel at appropriate times) you can buy Standard Saver Return for �85.90.
To check the off-peak pre-booked single fares, I simply selected a suitable 'quiet' day. (I chose the Tuesday after next, 21st November). All direct departures, from Derby to Edinburgh, up until 0942 are only available at full fare (�69.50). The 1042 and 1142 trains are both bookable for �15. A seat on the 1242 service is slightly dearer, at �21. The price drops back to �15 for the 1342 train but from 1442 onwards the price reverts to �69.50. (I've not checked all departures from 1442 onwards but that fare certainly applies to the first few trains).
I've done a quick check on a few other dates (and also on trains in the opposite direction). With a bit of flexibility, it's not difficult to find single fares of either �15 or �21in either direction.
Chris