A friendly warning, the following is bit esoteric, (a fancy word for hazy), and is still very much a work in progress. I'm having fun with it though, and I think it's an insightful review of the Taylor series in any event, so if you're interested, read on. My previous post was about a new way to memorize the Taylor series. It emphasized what the Taylor series did rather than how it was derived. I hadn't completely thought things through and had introduced a new term, "the inverse chain rule". +John Baez pointed out that the explanation as it stood sounded rather mysterious, and he was right. His comment made me rethink the entire Taylor series process again, and I think I, (hopefully), have a much more clear explanation now! My premise is that the easy way to memorize the Taylor series is to understand what is going on in each term. Hopefully, the same simple thing will be happening in each term making the whole process easy to remember and...