I have the GT Palomar 2001 edition mountain bike and the derailleur broke last year. I am now looking to fix it but I do not know which one to get as i do not want to do anything to the chain. Please can I have some help! thanks
Looking up the year, make and model of your bike on Bicycle Blue Book doesn't yield the make of the rear derailleur, but one can see it's a seven speed in the rear. Thus, you would want an RD compatible for that number of cogs. Since it appears the stock shifter on your bike is from SRAM, you'll want to get a SRAM rear derailleur. Whether the derailleur is marketed as 7,8, or 9 speeds doesn't matter as all these "speeds" of RD will work correctly with your shifter. If it happens that you have Shimano shifters, you'll need to stick with a Shimano rear derailleur. The same thing goes for the marketed speeds of a Shimano RD: 7, 8, or 9 "speed" all work the same for a 7, 8 or 9 speed Shimano shifter. (Front derailleurs are different. You can mix brands of shifter and derailleur with normal function. Bikes with a "mountain mix" of components will often have SRAM shifters and Shimano front derailleur's).
Here's some examples of derailleur's that will work for you (not a product endorsement, just examples): If you have SRAM shifters: SRAM X3 RD
If you have Shimano shifters: Shimano Tourney RD.
Almost all rear derailleurs can have their cages opened up by removing the bolt that serves as the axle for the lower tension pulley wheel. Separate the two halves of the cage by rotating them apart around the still attached upper jockey wheel axle. Now you can remove the derailleur fixing bolt of the broken derailleur and take the whole unit off the chain while the chain remains on the front chainwheel and rear cassette. I find it best when attaching the new derailleur (also with it's cage separated as above), to correctly route the chain, close the cage, replacing the lower tension wheel, then secure the derailleur with it's fixing bolt. Route the shift cable last and secure with the pinch bolt.