Two young mathematicians discuss the chain rule.
- 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 ) I can drive, without filling up, is miles, where is my car’s fuel efficiency, in miles per gallon.
- Riley
- Sure. So if you’re getting miles per gallon, that is 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 depends on , but depends on your velocity . That means is really a function of .
- Devyn
- Exactly! Finding the derivative is easy, but how do I find ?
- Riley
- Hmmm...