Understanding calculus formulas intuitively I am currently studying calculus in Russian and my course book is very rigorous.I used to think that I understand everything but I recently noticed that I only understand the logical steps in proofs of theorems and I actually don't understand all the formulas and theorems intuitively and can't see the motivation in proofs.Is there any way to improve intuitive understanding and really feel how mathematics works ,could you recommend any books?
 A: Look for the real world idea the proof is trying to capture; in calculus real world examples should be fairly reliable in many cases. Once you have this, intuitive hand-wavy reasoning can tell you the direction in which to apply your more formal efforts. For example, the product rule can be seen as follows:
Take a rectangle whose side lengths change through time. Let one side be given by $f(t)$ and the other by $g(t)$. How does the area $A=f(t)g(t)$ change? Extend two perpendicular sides in one direction (to represent changes in $f$ and $g$) to make a slightly larger box, and label the resulting L-shaped area $\Delta A$. This can be split into three rectangles, and we see that
$$\Delta (fg)=f\Delta g+g\Delta f+\Delta f\Delta g$$
Dividing by $\Delta t$ and (this is very handwavy) taking the limit as all the deltas go to zero gives
$$(fg)'=fg'+gf'$$
Is this a proof? Absolutely not. It glosses over a lot, and it only handles positive $f$ and $g$, as well as positive changes. But it gives you a hint of what you should be looking for. You could then take this result of how an intuitive example of product changes and create the formal setup:
$$\lim_{h\to 0}\frac{(f(x+h)-f(x))(g(x+h)-g(x))}h$$
And ask how to get this into something that looks roughly like
$$\lim_{h\to 0} f(x)\frac{g(x+h)-g(x)}h+g(x)\frac{f(x+h)-f(x)}h$$
Which is a natural first guess for the formal form of the sloppy area answer. In fact this doesn't quite work: what ends up working is replacing $f(x)$ with $\frac{f(x+h)+f(x)}2$ and similarly for $g(x)$. But the point is that we know where we want to go, and this makes our lives much easier. We might not get there immediately; how to get there can still be confusing. But this is an amazing help. If you're looking for motivation of proofs, look for the main idea that's being captured by the result of the proof and try to set that up to get an idea of where you're headed without worrying too much about formality. Often times part of a proof will suggest itself, and the suggestion can then be tackled properly. Directionless math is very rarely insightful.
A: Here is a nice book "The Calculus Gallery, Masterpieces from Newton to Lebesgue" by "William Dunham". You will find this book helpful for understanding many concepts. It is not an expensive book.
A: I don't have any book recommendations for you. I don't know how helpful reading a book is at gaining intuition in general. Of course a book can teach you material and show you a lot of interesting aspects to something. A book can also give you the background and tell you about ways to think about something, but in my opinion the way you really gain understanding and get a good intuition for something is by doing it.
So if you want to gain an intuition for calculus, I suggest simply doing a lot of calculus problems. When you have found $1000$ integrals, then you will automatically (IMO) being to see why this is working and it will become easier to solve/find integrals that you haven't encountered before. So maybe you real question is whether someone can recommend a good book with calculus problems. 
A: Here is a very nice book: Я. Б. Зельдович, И. М. Яглом, Высшая математика для начинающих физиков и техников ( the link:http://ilib.mccme.ru/djvu/zeld-yag.htm)
 This book is a priceless pearl. You can find it in Russian and English. Good luck.
A: Фихтенгольц is notorious in that respect. If you insist on studying in Russian, let me recommend a book by Грэнвиль В., Лузин Н. (same title, of course).
A: My free book Intuitive Infinitesimal Calculus emphasises intuition above all else, formalism and rigour be damned.
A: I will recommend Introduction to Calculus and Analysis by Courant. It's both rigorous and develops intuition. 
http://www.amazon.com/dp/354065058X
