Open the cover up and check for metal shavings on the magnet. Watch to see if the oil comes out silver or metallic colored. Check for play in the pinion by trying to wiggle the yoke. If youve got a bad bearing, then it is probably on the pinion. Seems to be more common there. How loud is the whine? It could be bad gear wear. For the price of a new set of gears plus installation, it would be cheaper to just swap in a different rear.
If it is bad, and its just your DD then Id suggest just swapping in a junkyard axle. If you were to go that route, open it up and check everything out the same way.