The passenger is not using a through train.
According to Rail Miles the shortest route is out of St. Pancras, as 124 miles 49 chains. The shortest route from Euston is 127 miles 60 chains. Therefore, the 3 mile rule can not apply.
We therefore, referto the National Routeing Guide, available here:
http://www.atoc.org/about-atoc/rail-settlement-plan/routeing-guide
All London Terminals (apart from Vauxhall - but that is not relevent here) are members of the London Group.
http://www.atoc.org/clientfiles/File/routeing_point_identifier.pdf page 38
Burton on Trent is a related station to the Derby Group, Lichfield Trent Valley, the Nottingham Group and Tamworth.
http://www.atoc.org/clientfiles/File/routeing_point_identifier.pdf page 11
London-Burton single fares are as follows:
SVS £61.80
SOS £87.50
London-Derby single fares are as follows:
SVS £86.00
SOS £89.00
Both these fares are more expensive than the equivalent Burton fare, so Derby is not an appropriate routeing point.
London-Lichfield TV single fares are as follows:
SVS £46.00
SOS £81.00
Both these fares are cheaper than the equivalent Burton fare, so Lichfield TV is an appropriate routeing point.
London-Nottingham single fares are as follows:
SOS £74.50
This is cheaper than the equivalent Burton fare, so Nottingham is an appropriate routeing point.
London-Tamworth single fares are as follows:
SVS £46.00
SOS £81.00
These are both cheaper than the equivalent any permitted fare.
Routeing permissions for London Group-Lichfield Group are:
TV, TV+WM
http://www.atoc.org/clientfiles/File/permitted_route_identifier.pdf page 603
Using these map combinations, it is impossibe to trace a route from Euston-Lichfield via Derby.
http://www.atoc.org/clientfiles/File/Maps.pdf pages 109 & 118
The shortest route from Lichfield-Burton is via Tamworth (unless something does Lichfield-Burton direct these days). Either way, you do not get the desired result this way.
Routeing permissions for London Group-Nottingham Group are: ER, ER+NL, NM
http://www.atoc.org/clientfiles/File/permitted_route_identifier.pdf
None of these maps provide a route out of Euston
http://www.atoc.org/clientfiles/File/Maps.pdf pages 38, 83, 84
Routeing permissions for London Grouo-Tamworth are: TV, TV+WM
http://www.atoc.org/clientfiles/File/permitted_route_identifier.pdf page 605
Like the Lichfield case, none of these maps provide a route from Euston-Tamworth via Derby, neither is Derby on the shortest route from Tamworth-Burton/
Therefore, I conclude that in my opinion, a London Terminals-Burton on Trent route Derby ticket is not valid out of Euston. If a passenger wishes to use such a ticket out of Euston, they should pay the appropriate excess fare to an any permitted ticket.