Translating intuition into rigor. The chain rule. When considering two functions $f(x)$ and $g(x)$, it is known that
$$\left(f\circ g(x)\right)' = f'\circ g(x)\cdot g'(x)$$
So my intuitive approach is:
$$\mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {g\left( {x + \Delta x} \right)} \right) - f\left( {g\left( x \right)} \right)}}{{\Delta x}}$$
$$\mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {g\left( {x + \Delta x} \right)} \right) - f\left( {g\left( x \right)} \right)}}{{g\left( {x + \Delta x} \right) - g\left( x \right)}}\frac{{g\left( {x + \Delta x} \right) - g\left( x \right)}}{{\Delta x}}$$
Put $g\left( {x + \Delta x} \right) - g\left( x \right) = \Delta g\left( x \right)$
$$\mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {g\left( x \right) + \Delta g\left( x \right)} \right) - f\left( {g\left( x \right)} \right)}}{{\Delta g\left( x \right)}}\frac{{g\left( {x + \Delta x} \right) - g\left( x \right)}}{{\Delta x}}$$
So I guess the problem boils down to translating how $\Delta x \to 0 \Rightarrow \Delta g\left( x \right) \to 0$ and to adress ${\Delta g\left( x \right)}$'s behaviour.
The last intuition is to recklessly write 
$$g\left( {x + \Delta x} \right) - g\left( x \right) = \Delta g$$
and put
$$\mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta f\left( {g\left( x \right)} \right)}}{{\Delta g\left( x \right)}}\frac{{\Delta g\left( x \right)}}{{\Delta x}}$$
which is the idea behind
$$\frac{{df}}{{dx}} = \frac{{df}}{{dg}}\frac{{dg}}{{dx}}$$

The chain rule is very simple, if you use the correct definition of the derivative. The derivative $f'(x)$ is a function such that
$$f(x + \epsilon) = f(x) + \epsilon f'(x) + o(\epsilon).$$
If you don't know what the "little oh" notation mean, think of it as
$$f(x + \epsilon) \approx f(x) + \epsilon f'(x).$$
Similarly,
$$g(x + \epsilon) \approx g(x) + \epsilon g'(x).$$
Therefore, using continuity,
$$f(g(x+\epsilon)) \approx f(g(x) + \epsilon g'(x)) \approx f(g(x)) + \epsilon g'(x) f'(g(x)).$$
We get the chain rule:
$$(f \circ g)'(x) = f'(g(x)) g'(x).$$
The only non-trivial part is
$$y \approx z \Longrightarrow f(y) \approx f(z), $$
which is a statement of continuity.
