How Derivative

Once in a while you come across a concept that is so sound and fundamental that it changes your entire outlook on the world. One such concept for me is something I learned in Calculus class fifteen years ago — the Derivative.
The derivative can be explained several ways. One way is to say that it is a measure of the rate of change. For example, interest rates are derivatives. So is inflation. Both of those are the rate of change of money with respect to time.
Another way to think of a derivative is to use a graph. The derivative is the slope of a curve at a particular point. If the curve is like a hillside, then the steepness of the hill at the point you are standing is the derivative. This is also consistent with our first definition, because it is the rate of change of the elevation of the ground with respect to position across.
Once you start looking for derivatives you see them all over the place. Gas milage — rate of miles you travel per gallon of gas. Gas prices — rate of dollars you pay per gallon of gas. Some disciplines are so used to derivatives that they don’t even come out and admit it. Financial people are notorious for this — the term “year over year growth” actually reflects the how much the rate of change of money the company makes per year is increasing. This is a rate of change of a rate of change. “Growth” is normally the rate at which something tangible increases (as in your height), but in financial circles, “growth” means the rate of the rate, or is actually a second order derivative.
(Second order derivatives can be described in several ways. One way that I like– No, this won’t be on the test I’m sharing it just because I like it. One way that I like is to think of a positive second order derivitave as a valley floor. As you hike down it gets less and less steep and eventually you’re heading upwards. A negative second derivative is like a mountain top, as you hike towards it it gets less and less steep and eventually you’re heading down. Some people also say a second derivative is like a smile when it’s positive (start out going down and then go up on the other side) or like a frown when it’s negative.)
I suppose I should come up with a use for a derivative now that I’ve explained it to you. Suppose you wanted to calculate the gas milage of your car. One way would be to create a table and record your odometer reading and how much gas you put in. Then you subtract adjacent odometer readings to see how far the previous tank of gas went. You can then divide the number of miles by how much gas you put in. Finally, average all the values.

Or you could use the Derivative. To do this, simply plot the odometer reading on the Y axis, and the total number of gallons you’ve put in the car on the X axis. Take a ruler and draw a line connecting all the points as best you can. The slope of that line (how many miles the line goes “up” divided how many gallons of gas the line goes “over”) is your gas milage.