As I started to use derivatives on daily base at college I wondered about one thing: Are you ( you - who is a person which is confident in using maths and using derivatives) able to proof the rules of derivatives without any help at any time ? Like when someone would come up to you this very moment and say "Here's a pen and a paper, write the proof for product rule, chain rule, (...) down" would you be able to do this?
The reason I ask: I am confident in using derivatives and work with the rules. I can show you what a derivative is off the cuff. I have done and understand the proofs of all the rules. But if someone would ask me "why does the chain rule work?" Well I had to look it up and that is what I wondered about:
Is it a usual thing in learning and using mathematics that you don't always have the proof in the back of your mind ?