If you have a written lease, then that is what controls. If the due date is the first, it is the first. If there is a grace period until the tenth, there is a grace period. The lease also likely dictates on what kind of notice a termination can be made, and if there is a cure for non-payment. Should the lease to a local attorney for more specific advice.
To questioners from West Virginia & New York: Although I am licensed to practice in your state (in WV, on inactive status as of 9/13), I practice on a day-to-day basis in Massachusetts. I answer questions in your state in areas of the law in which I practice, and in which I feel comfortable trying to offer you assistance based on my knowledge of specific statutes in your state and/or general principles applicable in all states. It is always best, however, to work with attorneys and court personnel in your own area to deal with specific problems and factual situations.