Differential Calculus Foundations and Differentiation Rules

Evaluate limits algebraically and numerically, analyze continuity, determine all types of asymptotes, and resolve indeterminate forms using L'Hopital's rule.

Compute first and second derivatives of polynomial, rational, exponential, logarithmic, and trigonometric functions using analytical rules.

Apply first and second derivatives to analyze monotonicity, concavity, points of inflection, graph behavior, and solve real-world optimization problems.

Apply linearity properties and basic integration rules to evaluate indefinite integrals for polynomials, 1/x, e^x, sin(x), and cos(x), and solve initial value problems.