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 ?

  • $\begingroup$ it depends on your education level, you will learn how to proof derivative rules in your second or third year of study as a mathematics undergraduate, at that time, may be you can prove them. Also, those rules are not that hard to prove. $\endgroup$ – Belive Oct 26 '17 at 9:28
  • $\begingroup$ I typically find there's anything from a one to three year delay in between understanding a body of theory, in the sense of having a big picture in your head of how it all fits together, what the theorems mean and how to use them, and going back and reading all the proofs of the fundamentals (and having them mean something, and not just be so much ink that my retinas scanned and forgot the moment I turned the page). $\endgroup$ – Jack M Oct 26 '17 at 9:31
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    $\begingroup$ I remember that the derivation of the product rule the teacher gave us had an "addition of the zero"-trick that was not quite easy to find. However, if we already know the product rule, it is in fact not hard using the differential-quotient-definition to prove that it is correct. $\endgroup$ – Peter Oct 26 '17 at 9:34
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    $\begingroup$ Yes I would. But the facts that you use daily when doing maths easily fill a whole book. You don't need to know all the proofs by heart but get a taste of them. IMO, if you tried to completely master the theorems involved in the proof of Last Fermat's theorem by Andrew Wiles, your whole lifetime would be quite insufficient. $\endgroup$ – Yves Daoust Oct 26 '17 at 9:34
  • $\begingroup$ To get that clear: I have worked through a few proofs for all these rules. But when you use derivatives all the time you only use the rule. And over the time I find myself in a place where I have to admit "Well I know how it works and have an idea why but let's be honest I have to look up the proof. I can not do it off the cuff" $\endgroup$ – LurioTabasco Oct 26 '17 at 10:06

There are difficult proofs that you may forget sometimes - that is nothing unusual. However, this is also a test whether you've really understood a proof. There's a huge difference whether you can derive it on your own or whether you only understand the steps of the proof.

I for one always try to understand the motivation and connections made in proofs - but of course, I do not always succeed.


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