Differentiation
Differentiation is the largest calculus domain in SPM Additional Mathematics. It tests the derivative concept, the standard rules, tangents and normals, stationary points and their classification, rates of change and small increments.
Rules
d/dx (xⁿ) = nxⁿ⁻¹ and d/dx (axⁿ) = anxⁿ⁻¹. Product rule: (uv)′ = u′v + uv′. Quotient rule: (u/v)′ = (u′v − uv′)/v². Chain rule: dy/dx = dy/du × du/dx. The tangent gradient at x = a is dy/dx evaluated at a; the normal gradient is −1/m. At a stationary point dy/dx = 0 and the second derivative classifies it: negative means maximum, positive means minimum.
Worked example
y = x³ − 3x² + 2. Then dy/dx = 3x² − 6x = 3x(x−2), so stationary points occur at x = 0 (y = 2) and x = 2 (y = −2). Also d²y/dx² = 6x − 6, which is −6 at x = 0 (maximum) and 6 at x = 2 (minimum).
Traps
Forgetting the inner derivative in the chain rule; using m instead of −1/m for the normal; assuming d²y/dx² = 0 settles the nature of a stationary point; confusing dy/dx with the value of y.