I read this article in high school, and it permanently changed my math life. Without it, I would continue to think I know derivatives because I learned twenty formulas and did forty exercises (which in fact every idiot could do).

I'll allow myself to give some advice for those who are interested in problem-solving, but have no experience. If you have some, then this should be well known.

* Keep problem / exercise ratio as high as possible. This is impossible with many calculus books; find a book with hard problems. An "exercise" is something which checks your understanding of definitions and theorems; a "problem" is something which exercises your skill and forces to think. Do exercises if the theory is unclear. If a task starts with "using mathematical induction prove that..." then it is an exercise. A problem forces you to think how to do it.

* Doing differentiation exercises will give you some speed, but after five-twenty minutes your brain will stop thinking and start to rot. Healthy mathematics - just like programming - hates doing the same thing again. Of course you have to learn some algorithms, but this is a tip of the iceberg.

* Always take 20 minutes (some say more) on a problem, unless you think it is ill-posed; giving up early is stupid. If you think the problem is impossible, try proving it. Think about some way of solving, reject it quickly if you made a thinko; if you sense "this might work" go deeper. Use paper.

* Don't read too much on philosophy of mathematics or biographies; this isn't deep from the mathematical side. Other articles (http://www.artofproblemsolving.com/Resources/AoPS_R_Articles...) are also worth reading. Check their forum (http://www.artofproblemsolving.com/Forum/index.php or www.mathlinks.ro).