This paper provides an accessible introduction to the concept of derivatives in mathematics. Beginning with an intuitive explanation of derivatives as a measure of how quickly something changes, the paper establishes essential vocabulary for describing positive, negative, and zero rates of change. It then connects derivatives to algebraic concepts — particularly graphing and slope — to show how the derivative represents the instantaneous slope at a given point on a line. The paper concludes by noting the broad applicability of derivatives in calculus for studying change across time, altitude, pressure, and other variables.
A 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 went on to specify 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, one could say that the derivative of the national debt was a high, negative derivative.
When working with derivatives, it is important to avoid ambiguity. While most people would assume that a "high" derivative is positive, the word "high" is not mathematically defined. For that reason, a specific 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 raises a natural question: what if the derivative is zero? If a derivative is the answer to the question "how quickly does it change?" and the answer is zero, that must mean it did not change at all. Therefore, if one were to say that the national debt was stable — that is, not changing — 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 in algebra is a graph — a system that plots points based on their values. Each point has two values, labeled X and Y respectively. The point is located X units to the right (if positive) or left (if negative) of the origin, and Y units above (if positive) or below (if negative) of the origin. The origin is defined as the point (0, 0), meaning "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 from left to right, one can see that the location of the object — itself a point — is changing as it moves along the line. Specifically, the X values are increasing (because the object moves from left to right), and the Y values are doing whatever the line is doing. If the line rises, the Y values are getting more and more positive.
"Derivative as instantaneous slope"
"Broad uses of derivatives in calculus"
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