When I reply to a mail from my boss, can I use "Noted with thanks "? Isn't it rude to my boss?

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It's certainly not rude. Your boss would appreciate it. However, it sounds slightly not natural. Noted. Thanks! Or Noted. Thanks very much. sound more natural. noted with thanks is perfectly understandable but not how a native would generally respond. When I read these questions I sure wish I could give more explanation as to why something sounds more natural than something else. Sometimes it is very hard to do. Your statement was perfectly correct, but a bit flowery. Noted. when used as a response to a request by someone, is a very short and succinct way to say that you understood what you were asked to do. It is like saying. I understand (and I will act do what you want me to do). But it is very short and not at all flowery. Attaching with thanks to it makes it sound a bit awkward because Noted is usually said alone. "Noted." That's why it sounds more natural to say. Noted. Thanks. "With thanks" sounds a bit more formal and like it should go with something a bit less abrupt than "noted!"