
Two young mathematicians discuss the chain rule.

Check out this dialogue between two calculus students (based on a true story):
Devyn
Riley! Something is bothering me.
Riley
What is it?
Devyn
It’s about rates of change.
Riley
That’s just a derivative. What’s the issue?
Devyn
It might take a bit of setup. Suppose my car has a 10 gallon tank. The distance (call it $D$) I can drive, without filling up, is $D(m)=10m$ miles, where $m$ is my car’s fuel efficiency, in miles per gallon.
Riley
Sure. So if you’re getting $35$ miles per gallon, that is $D(35) = 350$ miles, so you can drive 350 miles before running out of gas.
Devyn
Right! Now, the fuel efficiency depends on how fast I’m driving. If I’m driving 55 miles per hour, I can get 40 miles per gallon, but if I’m driving 70 miles per hour on the interstate, I only get 30 miles per gallon.
Riley
Oh! So your distance function $D$ depends on $m$, but $m$ depends on your velocity $v$. That means $D$ is really a function of $v$.
Devyn
Exactly! Finding the derivative $\frac {dD}{dm}$ is easy, but how do I find $\frac {dD}{dv}$?
Riley
Hmmm...
What is $\frac {dD}{dm}$?
Suppose that the fuel efficiency $m$ is a linear between $v = 50$ mph to $v= 70$ mph. What is $\frac {dm}{dv}$?
$10$ $-10$ $-1.5$ $1.5$