Don't both of them mean "Please do not be loud"?
What's the うるさく? Is it うるさい and change い into く to make it a noun? Could you guys suggest the topic that I can look into and study more about them?
Thank you so much.
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うるさくしないでください is the only correct choice. うるさくないでください is just plain ungrammatical because it lacks a verb.
～ないでください ("Please don't ～") of course needs a verb before it, and in this case, the verb is する (in its nai-form, し(ない)). うるさく is the adverbial form (aka "ku-form") of the i-adjective うるさい. うるさく is adverbially modifying する.