Advanced Integration Techniques: Substitution, Parts, and Partial Fractions

Derive the derivative definition from difference quotients, prove standard derivatives from first principles, and analyze second derivatives and curvature.

Differentiate exponential, logarithmic, and trigonometric functions using chain, product, and quotient rules.

Differentiate implicit equations and parametric curves, and solve connected rates of change modeling problems.

Construct tangent and normal lines, classify local and global extrema, and sketch complex transcendental curves.

Interpret definite integrals as limits of Riemann sums, apply the Fundamental Theorem of Calculus, and calculate areas under and between curves.

Evaluate non-standard integrals using algebraic substitution, integration by parts, and partial fraction decompositions.

Formulate, solve, and interpret separable first-order differential equations in physical, biological, and kinematic contexts.

Locate roots via sign change, solve equations using fixed-point iteration and Newton-Raphson, and approximate integrals with the trapezium rule.