Introduction to Derivatives

Business Calculus · Axiom Academy

LESSON Introduction to Derivatives The mathematical tool for measuring instantaneous rate of change In business, we constantly ask questions about rates of change : "How fast are sales growing right now ?" "What's the additional cost of producing one more unit?" "At what rate is revenue changing as we adjust price?" "How quickly is market share declining?" These questions all ask for the instantaneous rate of change —how fast something is changing at a specific moment. The derivative is the mathematical tool that answers these questions. Average rate of change tells you what happened over an interval. But business decisions often require knowing what's happening right now —not over the last month, but at this instant. You already know how to compute average rate of change : This is the slope of the secant line connecting two points on the curve. The derivative is what happens when we take the limit as the interval shrinks to zero—we get the slope at a single point! Watch what happens to the secant line as the second point approaches the first: As h → 0, the secant line becomes the tangent line , and its slope is the derivative! For f(x) = x², at x = 2, the derivative is exactly 4. This is the slope of the tangent line at that point. The derivative of f at x, written f'(x), is: Alternative notations for the derivative: Read: "dy dx" or "the derivative of y with respect to x" Example: Computing from Definition So if f(x) = x², then f'(x) = 2x. At x = 2, f'(2) = 4.

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