Mathematics: Derivatives Derivatives: An Explanation

Mathematics: Derivatives

Derivatives: an Explanation

Derivative" is a mathematical answer to the question, "how quickly does it change?" For instance, if one noted that the national debt was changing rather quickly, one could also say that the national debt had a high derivative. If one specified and went on to say that the national debt was rising rather quickly, one could also say that the national debt had a high, positive derivative. It follows that if the national debt were falling rather quickly (although that is unlikely to happen), one could also say that the derivative of the national debt had a high, negative derivative.

When working with derivatives, it is important to avoid ambiguity. While most would assume that a high derivate was positive, the word "high" is not mathematically defined. For that reason, a certain vocabulary should be used when working with derivatives to ensure effective communication. The words "high" and "low" should be discarded in favor of well-defined terms like "negative" (below zero) and "positive" (above zero).

Establishing that vocabulary begs the question: what if the derivative is zero? If a derivate is the answer to the question, "how quickly does it change," and the answer is zero, that must mean it didn't change at all. Therefore, if one were to say that the national debt was stable, or not changing, then one could also say that it had a derivative of zero.

Using some basic concepts from algebra, another definition for "derivative" can be reached. A common tool is algebra is a graph, a system that plots points based on their value. Each point has two values, labeled "X" and "Y" respectively, and the point is located "X" units to the right (if positive) or left (if negative) of the origin (the middle), and "Y" units above (if positive) or below (if negative) the origin (the middle). The origin is defined as the point (0, 0), or "X is zero and Y is zero."

If one plots two points on a graph and draws a line between them, then imagines an object following the line starting on the left and going to the right, one can see that the location of the object, which is also a point, is changing as it follows the line. Namely, the X values are increasing (because the object is going from the left to the right), and the Y values are doing whatever the line is doing. In the example shown, the Y values are increasing, because the numbers are getting higher and higher -- that is, according to the proper vocabulary, they are getting more and more positive.