Chain Rule Intuition We know that the chain rule is used to differentiate  a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or the other way around.
$$\frac{\mathrm{dy} }{\mathrm{d} x} = \frac{\mathrm{dy} }{\mathrm{d} u} \cdot \frac{\mathrm{du} }{\mathrm{d} x}$$
Despite we know that the above expression is not a fraction (even though it's a fractional notation of the derivative used by Leibnitz) you can "cancel" the two du's and get back dy/dx.
My question is: How can you even think of cancelling du from dy/du and du from du/dx when they are not even fractions. Just because it's been multiplied do they automatically become fractions?Are they really being "multiplied"?
I'am really looking for an intuition behind this.To me this is some kind of fantasy.It doesn't appear to be real.

In a race, Usain Bolt is travelling twice as fast as a train which is going 3 times as fast as a horse. How much faster is Usain Bolt travelling than the horse? 
$$ \frac{d\text{Bolt}}{d\text{Horse}}= \frac{d\text{Bolt}}{d\text{Train}} \cdot \frac{d\text{Train}}{d\text{Horse}} = 2\cdot 3 = 6 $$
