Is it possible to have intuitive explanations of rules like the product/quotient rule? I know it seems like a weird question, as these rules are simply a direct application of the limit definition of the derivative of a function.
What I am saying is, could these rules be explained somehow by using intuitive examples and analogies? For example, in my calculus textbook, in the section about the chain rule, the author gives the following note: 
"The chain rule should make sense intuitively as follows. We think of dy/dx as the (instantaneous) rate of change of y w.r.t x, dy/du as the rate of change of y w.r.t u & du/dx as the rate of change of u w.r.t x. So, if dy/du=2 (which means that y is changing at twice the rate of u) & du/dx=5 (u is changing at 5 times the rate of x), then it should make sense that y is changing at 2*5=10 times the rate of change of x, or dy/dx=10). 
Of course he is refering to the expression (dy/dx)= (dy/du)*(du/dx).
So, is it even appropriate/logical to ask such a question, about the intuitive explanation of these particular rules or rules similar to them? & if not, why? 
I'd just like to note something, I am completely convinced that the rule is correct and that it works, because we can just check and confirm it quite easily. And I do not have a problem with applying it. 
What I'm just wondering about is wether it can be explained in other ways than just the direct application of the definition of the derivative. 
 A: For the product rule, imagine having a rectangular window open on your computer, measuring $x$ by $y$. You drag the upper right corner of your window, increasing its size. You can divide the added on window area into 3 parts: a rectangle on the right that has area $y \Delta x $, a rectangle on top that has area $x \Delta y$, and a rectangle in the upper right corner with area $\Delta x \Delta y$. If the total change in area is small, that upper right rectangle contributes very little to the extra sum in area. So really, the total change in area is about $y\Delta x + x\Delta y$. If we think about this change in area as a function of time, and take a limit, we get the product rule.  
A: This was going to be just a comment, but it seems that the formatting doesn't look clear as a comment.
It's simple to discover the product rule and many other rules by using the basic approximation 
$$
f(x + \Delta x) \approx f(x) + f'(x) \Delta x.
$$  
If $h(x) = f(x) g(x)$, then 
\begin{align}
h(x + \Delta x) &= f(x + \Delta x) g(x + \Delta x) \\
&\approx (f(x) + f'(x) \Delta x) (g(x) + g'(x) \Delta x) \\
&= f(x) g(x) + (f'(x) g(x) + f(x) g'(x))\Delta x + f'(x) g'(x) \Delta x^2 \\ &\approx h(x) + (f'(x) g(x) + f(x) g'(x))\Delta x.
\end{align}
The product rule just pops out automatically.  You can even turn this into a rigorous proof.
