I purchased an RV during the tax year that I used personally for 22 days and rented out for 5 days. I understand that, under the 14 day rule, I do not have to report the income but I am not allowed to claim expenses. Am I allowed to claim the depreciation of a business asset for the RV under this rule?
No, you shouldn't take depreciation expense. Depreciation is just a way to take a rental expense for the rental structure, usually a house, but in this case a RV. When you aren't reporting income because the property was rented for less than 14-days, you can't report any expenses either, including depreciation.
If you have more going on with Business use, you can add more info in the comments.