One source of data for “which routes do cyclists use” is the Strava data collected for all the users of Strava. Whilst the data cannot be taken as typical of all cycle users in Derby, the results are interesting in showing some of the most popular routes. Note that the data is only up to May 2015.
For example, the Riverside Path is very well used as is Duffield Road and the A6 north of Derby.
You can see the Strava heat map data at this link The more heavily used routes are shown in red, the lesser used ones in yellow or green.