Differential Calculus: Advanced Techniques
IB Mathematics Analysis and Approaches (AA) HL Paper 2 and Paper 3 test differential calculus at a depth beyond SL, including implicit differentiation, related rates, and optimization in context. Implicit differentiation applies when y is defined implicitly by an equation rather than explicitly as y = f(x). Technique: differentiate both sides with respect to x, applying the chain rule whenever y appears: d/dx[y²] = 2y(dy/dx). For x² + y² = 25, differentiating both sides: 2x + 2y(dy/dx) = 0 → dy/dx = -x/y. This gives the slope of the tangent at any point on the circle without solving for y explicitly. Related rates problems connect rates of change of two quantities through a geometric or physical relationship. Standard IB HL related rates: a ladder sliding down a wall (connecting the rate at which the base slides out to the rate at which the top descends using the Pythagorean theorem), filling a conical tank (connecting volume change rate to height change rate using the cone volume formula V = (1/3)πr²h with r expressed in terms of h using similar triangles). Protocol for related rates: (1) Identify all variables and their rates of change; (2) Write an equation connecting the variables; (3) Differentiate implicitly with respect to time; (4) Substitute known values at the specific moment. Optimization: use the first and second derivative tests to classify critical points. The first derivative test: if dy/dx changes from positive to negative at x = c, then c is a local maximum; negative to positive → local minimum. The second derivative test: if f''(c) < 0, c is a local maximum; f''(c) > 0, local minimum; f''(c) = 0, test inconclusive.