In looking at what it takes to win a derby, let's start with what causes most people not to win. In my experience, about one-third of all cars retire due to mechanical breakdown, most of which could have been prevented with better preparation. Another 25% get their cars stuck for one reason or another, and the remaining 25% are legitimately eliminated by the other competitors (which leaves 20% of the entrants who get to go onto the feature heat).
Assuming you get your car in reasonably good mechanical condition, you should be able to concentrate purely on strategy during the race.
Naturally you want to avoid taking any big hits, and there are a couple of ways to do this. First, try to stay away from the corners of the ring where people can get a good run at you. This also has the secondary benefit of reducing the likelihood of your getting stuck up against the blocks. Second, as you drive around the ring, do so in a counter-clockwise direction whenever possible. This way your driver's door is facing the inside of the ring, and other drivers are not allowed to hit you in the driver's door.
If you see a hit coming that you just can't avoid, it's best to take it broadside on the passenger's side if at all possible. The reason is by process of elimination: you don't want it on the front where it could kill the engine; you don't want it in the rear because you have to save your rear for your own hitting; and you don't want it on the wheels where it could blow a tire or bust an axle.
Always try to keep moving, even when not lining up your next victim or avoiding being someone else's. If you get hit unexpectedly this will help dissipate the energy of the impact because a spinning tire will allow the car to slide with the hit much easier than a non-moving tire. Also, by keeping the revs up you reduce the chance of stalling your engine.
When making hits try to use your rear as much as possible to protect the drivetrain in the front. I usually make an exception to this rule since the fronts of my Caprices are always much stronger than the rears. In fact, I end up protecting the rear for fear of having the exposed rear axle hit.
I guess what I am trying to say is to protect the weakest part of your car, whatever that may be at that point in the race.
If You Get Stuck
You're screwed. However, don't just floor it hoping to power your way out, because you'll only dig yourself in deeper. You can try to gently rock back and forth, but don't gun it unless you know you are just about to be counted out. Quite often another car will hit you and this can help you free yourself, providing you haven't already dug yourself in.
Sandbagging is the practice of avoiding making any contact in the hope that your car will be the last one running. This is seldom tolerated by the derby officials, and of course it isn't fair to the other drivers. Besides, why wouldn't you want to make hits? That's why we enter derbies in the first place. There are few more exciting moments in motorsports than when you finally get someone lined up in your crosshairs and then nail them as hard as you can. If you are entering derbies just to make money, then you must be running richer derbies than I do. (I figure it costs me about $300-500 to enter a car, not including time spent. Winning your heat might pay about $200, and winning the feature pays $1000.) When I'm in the ring, the adrenaline is pumping so hard that as soon as I see a target, I go for it as hard as I can.