Chain Rule Intuition
Solution 1:
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?
$$ \frac{d\text{Bolt}}{d\text{Horse}}= \frac{d\text{Bolt}}{d\text{Train}} \cdot \frac{d\text{Train}}{d\text{Horse}} = 2\cdot 3 = 6 $$
Solution 2:
If $h$ and $g$ are linear functions, then it should be obvious what the chain rule must hold. The general chain rule is simply this observation plus the fact that derivatives provide good linear approximations.