How to write good Response Letters for the reviewers

If the reviewers are in error, yes you should show it to them, but you should often also take the opportunity to revise your paper, so that your readers don't fall into the same error. For example, suppose reviewers say that you could use such and such an approach to get a better result. If they are right, then you should either do it, or explain why you're not doing it.

But if the reviewers are wrong, then maybe you should address this idea in your introduction. In one paper I wrote, we included in our introduction a false one paragraph proof of the thing we took 10 pages to prove. Afterwards we pointed out the error.

For one paper we submitted, we got conflicting referee reports. Essentially, the referees both wanted us to revise the paper, but in different directions (to oversimplify: one reviewer said "shorten and simplify" and the other said "lengthen and elaborate"). We agreed with one report, and disagreed with the other. So in our response, we pointed out that we couldn't do both . We explained why we preferred the choice we were making, and our paper was accepted.

You don't have to do everything the reviewers suggest. But if you don't, your response letter should explain why you have chosen to do otherwise.