Another alternative way of teaching calculus is via infinitesimals (for example the book Elementary Calculus, An Infintesimal Approach by Keisler). The way of thinking about calculus via infinitesimals is obviously very natural, and mathematicians (e.g. the pioneers of calculus, Euler etc...) have used arguments using infinitesimals long before they should have been really allowed to do so. Keisler's book (and in general the area of Non-standard analysis') makes rigorous our intuition regarding infinitesimals, and is a set of rules that teach us how to formally reason with them. In my opinion this system is intuitive, but the student can never really have a proper understanding of what they are doing "from the ground up" with out some basic knowledge of model theory. The limit approach is less intuitive, but at least a student doesn't have to just accept some rules without truly understanding what's behind them. Possibly this infinitesimal approach is a half way house between teaching it properly with limits and just teaching rules of differentiation to people who aren't interested.