A-Level Mathematics: Differentiation Techniques: Chain, Product, Quotient, Parametric
A-Level Mathematics — Differentiation Techniques: Chain, Product, Quotient, Parametric: quick notes, common traps, and an exam-style example.
Chain rule
For y = f(g(x)), dy/dx = f′(g(x)) · g′(x). Always label ‘inside’ vs ‘outside’. Example: if y = (3x²+1)⁵, outside derivative = 5(… )⁴, inside derivative = 6x → dy/dx = 5(3x²+1)⁴·6x.
Product & quotient rules
Product: if y = uv then dy/dx = u'v + uv'. Quotient: if y = u/v then dy/dx = (u'v − uv') / v². Always bracket the numerator to avoid sign errors. In Cambridge A-Level scripts, lost minus signs = lost method marks.
Parametric differentiation
If x = x(t), y = y(t), then dy/dx = (dy/dt)/(dx/dt). After differentiating, substitute t back into x and y forms if the question wants dy/dx in terms of x and y. This is common in H2 Math application questions (t is time / angle).
Exam habits
- Write the rule first (“Using quotient rule: …”) before you start differentiating.
- Simplify only at the end — do not expand too early and invite algebra slips.
- State where dy/dx is undefined (denominator = 0) if the question asks about tangents / normals.