Using Database Engine Tuning Advisor in production
Don't go to DTA.  It can have severe performance impacts in production and can really do more harm than good (I learned that lesson the hard way when I started out as a DBA).

The first step is to find out what exactly is causing your performance issues.  Is it a CPU issue? Memory? Improper indexes? Also, how do you know you are experiencing a performance issue?  Are users complaining of slowness? Are you seeing CPU spike high and stay there?  What about memory? Is Page Life Expectancy dropping often and you can't explain why?  These are the questions you need to ask.  You need to find out where it hurts first, then go from there.

So, to answer your question, no you shouldn't use DTA [in production, during business hours].

EDIT:

This answer may help you as well: Should I rely on Database Engine Tuning Advisor for creating indexes?

The important thing to realize is that just because DTA recommends something, there is still ALOT to be said for using your knowledge of the DBs and data in conjunction with it.  Same goes for the wonderful Missing Index Warnings.  Can DTA get you started? Absolutely, but there's much more to consider when making changes that it recommends and the performance impact of running it in production.

I, personally, have used it less than 5 times in my career.  I try to use DMVs/queries in conjunction with our 2 monitoring solutions to troubleshoot performance issues.