Yes. You should use synthetic. Under load, that axle gets hot. Synthetic stands up to the heat better.
The tag on my differential cover says "Use Synth. Oil." You should use the weight of oil that your owner's guide specifies. For my 2002, it is 75W-140. I changed my oil 3 weeks ago at 100,000 although the owner's guide says I can go to 150,000.
The axle takes 7 pints. The stuff I used is about $12/qt.
Don't forget the limited slip additive if you have that kind of axle. Look at the codes on the differential tag or the door tag to determine the type of axle you have. Your owners manual will have the amount that you should add.
Be careful in using any other weight (viscosity) of oil than what is specified in your manual. Some of those other viscosity synthetics are ment to replace dino-oil in cars and trucks.
You'll also need a tube of some silicone gasket material. This is the sealer used for the differential cover. Make sure the surfaces are clean and oil free before using. Follow the directions. Helps to let the seal set for a few hours before adding oil.