If during revision process an author discovers and corrects errors not detected by reviewers, should the author inform the editor of these changes? I recently submitted my manuscript to Journal of optics (IOP Publishing). A couple of days ago, I received an email regarding initial decision on my paper. Thankfully, it is a "moderate revision". Referees also have made some comments about typos and grammar and a point or two about the core material of the paper which is only to clarify more. So far, it seems a possible acceptance for my paper. doesn't it?. 

The thing is, when I was making the amendments based on the referees' comments, I found that four of the formulas used in the paper are incomplete and I wrote them wrongly. But it does not affect the rest of the paper in any way.  The reviewers did not notice these errors.

This is my first paper. I am a worried that they might reject the manuscript because there were some errors in the original submission that I have now corrected.

I am obliged to make a point-by-point response to the reviewers' comments and make a list of the changes made to the paper. 

So, my questions are


Is it common for an author to detect additional errors in a paper during the process of making revisions? 
Should I just correct them and make no mention of them in my report to the editor?
This is a straightforward question with a simple answer: correct whatever you find is wrong, even if it has not been raised by the reviewers.

I have an experience of more than 600 publications (see Google Scholar for instance) and have reviewed even more manuscripts as peer reviewer, editorial board member or editor, and it is commonplace to have the need and opportunity for improvements in a manuscript during the revision process.

Be candid, disclose the correction. Possibly don't say it was a mistake in the first place, but present it in a positive fashion, saying it is a correction or improvement in the paper.

In most cases reviewers and editors will be pleased by your self-reviewing effort.