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For example, about the basic differential Calculus, I'm doing it in the following way, and is it the good method ?

I only focus and try to obtain the fully understanding and intuitive sense of the most basic and fundamental concepts or theorem, but for others things which could be proved or just 'special case aspect' of the basic concepts, I only to referring them when I use it, but not to try to understand or memorize.

Like, The concepts of Limits and Derivative and Theorem of Differentiability implies Continuity are the three most fundamental stuffs. I'll put focus on that to fully understand why it's true.

But for the rules, like $f'(x^n)=nx^{n-1}$, or sum rule, product rule, quotient rule and so on, I only know they are proved from the basic concept of derivative, i.e. $\lim_{\Delta x\rightarrow 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}$. All the rules are from this basic concept. So I understand this basic concept intuitively means we find the instantaneously changing rates with respect to $x$.

But for the rules, like integer power differentiation, i.e. $f'(x^n)=nx^{n-1}$, I cannot 'understand' it in the same way with that of basic concept of derivative(instantaneously changing rates with respect to $x$). Only thing I know is that it's proved from the definition of derivative.

So, is it enough for mastering the rules ? Because for definition of derivative, I could still graph a function, and to see the trending when $\Delta x \rightarrow 0$, the limit will get the instantaneous changing rate at the point of $x_0$. But for the proved rules, like $f'(x^n)=nx^{n-1}$, it's very hard to get the intuitively feeling in the same way with the derivative.

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It is important to have a good intuitive feel for the basic concepts. However, very soon you will be using the basic definition fairly infrequently. The "other rules" that you propose not to memorize will have to become as familiar and automatic as the basic methods of arithmetic and algebra. For most functions that you want to find the derivative of, going back to the definition will not be feasible. – André Nicolas Sep 28 '11 at 9:20
@André Nicolas , sometimes "other proved rules" are pretty hard to get the intuitively feeling in the same way with basic concepts,I don't know why. But is it okay to deal with these "other proved rules" in the way of 'practice makes perfect' only to use it familarly ? And only to know how to prove it & where is it from in order to get some kind of "safety" feeling ? – Xingdong Sep 28 '11 at 11:09
With something like the Quotient Rule for differentiation (you will see it soon), knowing where it comes from will not add much, but being able to use it automatically will be essential. With some other rules, like the Chain Rule, some intuition is possible and useful. By the way, there is a potentially bad error, affecting the understanding, in your line (involving $\Delta x$) about the definition of the derivative. There should be no $=$ sign. – André Nicolas Sep 28 '11 at 13:12
@André Nicolas ,Thank you,I've corrected error. And,sometimes it seems if I could get the intuitively feeling on the basic concepts, then I'll be unnecessary to memorize it, it'll be kinds of obvious, even though forgetting it, we can always 're-create' it on our own from the 'nature', you know, this kind of feeling. But it seems only most fundamental concept will work in this way, but I don't know why some many 'proved rules' or combination of basic concepts or so, are almost impossible to do that.I only know they're essentially from basic concepts, but still don't have that kind of feeling. – Xingdong Sep 28 '11 at 18:19
You are in principle right. Many students get into trouble because they have a good memory, and in school always found it more efficient to memorize. In the long run they know nothing. It can be much more efficient for the long run to know what's going on! However, basic algebraic proficiency is an exception, and this includes differentiation. It has to be automatic, like walking. – André Nicolas Sep 28 '11 at 18:56
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Learn the history of Analysis, Specially History of Analysis from 16th century to Euler and Cauchy, Weistrauss School and Cantor. Analysis by history and History Of mathematics by John Stillwell. That will instill the sense of motivation for the ideas.

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