Derivatives Summary

Business Calculus · Axiom Academy

SUMMARY The Derivative - Summary Key concepts and formulas from Unit 3 The derivative measures the instantaneous rate of change of a function. It tells you how fast a quantity is changing at any specific point. Geometrically, it represents the slope of the tangent line to a curve. Average vs. Instantaneous Rate Average rate of change measures change over an interval (slope of secant line). Instantaneous rate of change measures change at a single point (slope of tangent line). The derivative is the limit of the average rate as the interval shrinks to zero. This limit represents the slope of the tangent line at point x. Power Rule: For any term like x^n (including negative and fractional exponents) Product Rule: When two functions are multiplied and can't be easily expanded Quotient Rule: When one function is divided by another (both contain x) Chain Rule: For composite functions - a function inside another function Exponential & Logarithmic Derivatives The function e^x is unique: it equals its own derivative! This property makes it essential for modeling continuous growth and decay, including compound interest, population growth, and radioactive decay. If C(x) is total cost, then C'(x) is the marginal cost - the cost of producing one more unit. If R(x) is total revenue, then R'(x) is the marginal revenue - additional revenue from selling one more unit. If P(x) is profit, then P'(x) is marginal profit. Maximum profit occurs where P'(x) = 0.

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