Can't believe none of the Derby-ites have replied to you yet!
Not much really technical riding close to Derby - there is the BMX track if you want to brush up your skills at Alvaston Park.
Going north-ish from Derby, there are quite a few bridle paths out Quarndon/Kedleston way and up towards Ambergate/Belper/Chevin. Easy to ride to from the Uni end of town.
If you have a car or take the train, Matlock is only half an hour away, and you will get out towards Chatsworth on a decent ride from there, but lots of stuff around Ambergate/Cromford/Crich, all good apparently, some cheeky, though I don't ride or know that area.
South-ish the mag had some routes for Ticknall a while back, but I'm not sure what is actually there.
West-ish - towards Ashbourne/Subury is not great for riding off road, according to a mat ewho lived that way
East-ish there is a good range of cyclepaths and towpaths and Sustrans routes that can get you over to Ilkeston and into the heart of Nottingham traffic free. Great fun if you just want to spin your legs, if the trails are claggy or if you fancy an evening beer at a waterside pub.
Also Sustrans/cyclepath options can get you onto the High Peak and Tissington trails, neary forgot.
North-eastish takes you to the mountain biking mecca that is Ilkeston (no local bias on here). To my surprise - I'm not a local originally - there is a lot of decent bridleway bashing around here - we have a couple of regular local 20 mile loops, one towards Strelley and Nottingham, one around Breadsall and Drum Hill. We can link them up to make a big day out if we want. There were a few jumps at Drum Hill but they have been flattened.
Put a shout out when you get here and there are a lot of guys around who will be able to help in more detail or take you out if you want some guided highlights.