Chain Rule Intuition

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?

$$ \frac{d\text{Bolt}}{d\text{Horse}}= \frac{d\text{Bolt}}{d\text{Train}} \cdot \frac{d\text{Train}}{d\text{Horse}} = 2\cdot 3 = 6 $$

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.

Here's the intuition I give every time I teach the Chain Rule:

Remember that derivatives are rates, the Chain Rule explains how to meaningfully multiply these rates together. A cheetah is 4 times as fast as a man, and a man is 10 times as fast as a snail. You can see right away how to compare the cheetah to the snail-- the cheetah is 40 (that is, 4x10) times as fast.

The Chain Rule is just the formula for computing more difficult derivatives by using an intermediate step. We have $y=f(x)$, and we can get the rate of change of $y$ with respect to $x$ by going through an intermediate variable $u=g(x)$ (where $f(x)=h(g(x))$). We get $$f'(x)= h'(g(x)) \, g'(x)$$ or, equivalently, $$ \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}.$$

---

The above is just a quick intuitive explanation for why the Chain Rule involves multiplying derivatives and "canceling." Rahul's answer explains a proof for this fact.

Having proofs is essential, because sometimes derivatives may not work as you expect. For example, if you have $z=f(x,y)$ where $x$ and $y$ are both functions of $t$, the Chain Rule looks like $$\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt},$$ which is not the same as ordinary fraction cancelling.