The stipulation governs. If the stipulation is silent on this issue all the responding party must do to comply with it is serve it on the last day. If served by mail, which is the usual method, you will get it in due course.
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The exact deadline is governed by the stipulation. If the stipulation states that the response must be in the recipient's hand by certain date that provision governs. If the stipulation is silent on the method of service, I would assume that you may serve by any appropriate means by the due date -- including depositing the response in the mail. Since the Code normally provides 30 days to respond by any allowable means of service including by mail, a 15 day extension should mean you have an extra 15 days to respond by any proper means of service including depositing the response in the mail. If you want to be absolutely certain, get a clarification from the opposing side. Good luck.
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I would argue that "due to the defendant by the 15th" means that the defendant must have it on the fifteenth. I also think that most attorneys have too much shame to raise a stink about getting the discovery served on the 15th a day late.
Clarifying with opposing counsel is the best idea.