The driveshaft on a BMW is a length of metal connected to the gearbox at one end of the vehicle and to the differential at the other end. There are a couple of components that can fail on a BMW driveshaft, including the centre support bearing and the rubber flex disc (if it is a model equipped with one), which keep the driveshaft centred on the car and help control vibrations. If the centre support bearing goes out, it causes very loud thumps as the driveshaft moves around.
Raise your BMW with the floor jack and place it on jack stands. You can either raise one side of it or raise it at all four corners. Either way, make sure it is raised high enough to allow you to comfortably reach the driveshaft. Also, make sure that the car is safely supported since you will be working underneath. You can slide a couple of wheels and tires under as well for added safety.
Spray penetrating oil on all the bolts that are going to be removed and let the oil soak in (the longer the better). Disconnect any sensors that go from the gearbox to the driveshaft. Not all models will have sensors, but some newer models like the 750 have sensors that must be disconnected. Note where the sensors go for later reinstallation.
Unbolt the exhaust hangers from the exhaust brackets. Disconnect the manifold bolts so that the exhaust can be dropped out of the way. (Depending on the model, you may be able to just loosen the manifold bolts a little to allow the exhaust to drop down enough.) Support the exhaust system with something (jack stands, block of wood, etc.) if you are not removing it entirely.
Remove the heat shield located between the exhaust system and the underside of the car (attached with a variety of bolts, depending on the model). Set the shield aside.
Remove the transmission crossmember, if needed, in order to access the bolts that hold the driveshaft to the transmission. Support the transmission with something during this procedure.
Remove the bolts that hold the driveshaft to the transmission and the bolts that hold it to the differential. Remove the bolts that hold the centre support bearing in place. Pull the driveshaft out from under the car.
Mark the two halves of the driveshaft where they meet so that when they are put back together they are properly oriented.
Loosen the large nut behind the U-joint near the centre bearing a little, pull the driveshaft apart, then loosen the nut again, repeating until the driveshaft is separated.
Pull the centre support bearing off with a gear puller. Note the orientation of the bearing so that the new one is properly installed.
Install the new bearing by tapping it onto the driveshaft with a socket or similar tool that can fit over the splined part of the driveshaft. Tap the bearing on until the edge of the input shaft is just visible.
Reinstall the front shaft section, ensuring that the markings are properly lined up. Reinstall the driveshaft at both ends of the car in the reverse of removal. When you reinstall the driveshaft, preload the centre bearing 4 to 6mm toward the front of the car. Center the driveshaft in the transmission tunnel and tighten the bolts for the centre bearing.
Reinstall the transmission crossmember if it was removed. Reinstall the exhaust system. Lower the car to the ground.