Rates of Change

What, then, might the derivative be a model of?

It's not the model of a thing ...but rather of an idea.

It's not a toy store model, constructed from plastic and glue ...but rather a mathematical model, constructed from functions and their changes.

Let's squint a little harder...

Functions tell us how one thing depends on another: The unique  y  that corresponds to a particular  x . Functions are themselves models. They provide a mathematical description of things as various as:

• The amount of water in your bathtub
  
  
• The speed of your car as you brake
  
  
• The effects of calorie consumption on your weight

Or:

• World population
  
  
• The effect of pollution on marine organisms
  
  
• The speed of rotation of a galaxy so far from its center

Lots of things! Functions let us describe many cause-and-effect relationships in the world.

Of course, things change. The  y  that corresponds to a particular  x  today may not be the  y  that corresponds to that  x  tomorrow. Your bathtub fills up. Your car slows down. The more you eat, the heavier you become.

Often, we can't keep track of all the world's changing values. Instead, we summarize what we see by talking about the rate at which something is changing:

• Your bathtub is filling at a rate of one gallon every two seconds
  
  
• Your car slows down at a rate of 10 mph every second
  
  
• An extra 500 calories each day translates into an additional pound

It's more likely that we will know the rate at which something is changing than have a complete knowledge of the thing itself.

The derivative gives us a mathematical way to talk about rates: Anything described by "So much change in  y  per so much change in  x".

Don't get it? Don't worry. Derivatives are given a careful study in calculus, not precalculus. Precalculus will talk a lot about functions, how they change, and, indeed, their rates of change. However, if you understand how "rate" is used in everyday language, then you already understand how it is used in precalculus.

Just keep listening for the word: "rate", "rate", "rate". It's like listening to foreign speakers on an audio tape. By the time you get to calculus, the idea will be so second-nature that the careful definition of "derivative" will seem like only a clever turn of phrase.