Differentiation Rules for Complex Functions
HSC Mathematical Methods Year 12 requires fluency with three advanced differentiation rules beyond the basic power rule. Chain Rule: for composite functions f(g(x)), the derivative is f'(g(x)) × g'(x). Example: d/dx[sin(3x²)] = cos(3x²) × 6x = 6x cos(3x²). Identify the outer function (sin) and inner function (3x²), differentiate each, then multiply. Product Rule: for u(x) × v(x), the derivative is u'v + uv'. Example: d/dx[x²eˣ] = 2xeˣ + x²eˣ = xeˣ(2+x). Quotient Rule: for u(x)/v(x), the derivative is (u'v − uv')/v². Example: d/dx[sin x / x] = (x cos x − sin x)/x². NESA HSC exams frequently present 'disguised' composite functions where students must identify the correct rule before differentiating — common errors include treating (2x+1)⁵ as a product rather than a chain rule problem. Practise identifying the function type before applying any rule: ask 'Is this one function inside another? Two functions multiplied? Two functions divided?'