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Have you changed the differential oil yet? That is the cheapest and best first step in diagnosing the noise. Is/was the oil black with a strong smell, contain metal shavings(use a magnet to check), was the oil level correct prior to changing?
Locate a gearing specialty shop. They usually spend the majority of their time working with race cars, setting-up differentials, and drive-trains. Even experienced mechanics refer people to the gearing shop here, because properly setting the gears "backlash" is very critical, and if a mechanic doesn't do it often they lose their "touch" to be efficient. Improperly set, the gears will be very noisy, wear unevenly, cause premature bearing failure and may cause other drive-train problems.
If you are mechanically inclined, have the tools, initiative, and time, check out
ACCU Auto Parts technical article where they help you do it yourself.