Unit 2: Differentiation: Definition and Fundamental Properties

Unit 2 is where derivatives begin. You'll connect average rate of change (slope of a secant line) to instantaneous rate of change (slope of the tangent line), then formalize that idea with the limit definition of the derivative in both forms. Next you'll see how differentiability relates to continuity (differentiable implies continuous, but not the other way around: watch for corners, cusps, and vertical tangents). The second half is all about rules: the power, constant multiple, and sum rules, the derivatives of sin x, cos x, e^x, and ln x, and the product and quotient rules, finishing with tan x, sec x, csc x, and cot x. Recognizing a limit as a derivative in disguise is one of the most tested skills on the exam, so practice it until it's automatic.