The Derivative Definition

Business Calculus · Axiom Academy

LESSON The Derivative: Definition & Business Meaning Understanding instantaneous rate of change and what it means for business Suppose a company's revenue R(t) depends on time. We can easily find the average rate of change between two times: But what if we want to know the exact rate of change at a single instant ? How fast is revenue growing right now ? The secant line connects two points. As we move the second point closer to the first, the secant approaches the tangent line —and its slope is the instantaneous rate of change! The derivative of f(x) at x = a is: We take the average rate of change over an interval of width h , then shrink h to zero. Slope of the secant line between two points Slope of the tangent line at one point Use the slider to see how the secant line becomes the tangent line as h 0 : Secant slope: 4.0 | Tangent slope (derivative): 2.0 As h gets smaller, the secant line (blue) rotates toward the tangent line (purple). The secant slope approaches the derivative value. At h = 0 , they would be exactly equal! What Does the Derivative Mean in Business? In business contexts, the derivative tells you the rate of change of one quantity with respect to another. Here's how to interpret common derivatives: A factory's cost function is C(x) = 1000 + 50x - 0.1x^2 dollars for x units. The marginal cost is: C'(x) = 50 - 0.2x At x = 100 units: C'(100) = 50 - 20 = \ 30 Interpretation: When producing 100 units, the 101st unit costs approximately 30 to make.

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