Lesson 3: The Product Rule & Quotient Rule
Applying differentiation methods, rules, optimisation, and related rates for NCEA Level 3 Calculus.
🎯 Ngā Whāinga Akoranga | Learning Intentions
Product rule d/dx [u v] = u v' + v u' and Quotient rule d/dx [u / v] = (v u' - u v') / v^2 for composite product/fractional functions.
Identify component functions u(x) and v(x), differentiate product and quotient combinations, and simplify algebraic derivative expressions.
🎥 Media Anchor & Pedagogical Scaffold
Product Rule — Khan Academy
Video (2 min 39 sec, Khan Academy): Introduces the product rule — d/dx [f(x)·g(x)] = f′(x)g(x) + f(x)g′(x) — this lesson's core rule. Watch in full, then the lesson extends the same structure to the quotient rule.
🧠 1. Before Viewing (Activate & Predict)
Why is the derivative of a product NOT simply the product of the individual derivatives?
👁️ 2. During Viewing (Watch With a Job)
Watch the full 3¾-minute clip on the power rule. In your calculus logbook, record:
- The power rule formula: Write d/dx [x^n] = n x^(n-1) with three worked examples: d/dx [x³], d/dx [x^(-2)], and d/dx [x^(1/2)].
- Why it works: Note the geometric or limit-based reasoning the video uses to justify the rule.
- Application prep: Differentiate f(x) = 5x⁴ using the power rule. (Answer: f'(x) = 20x³.)
🗣️ 3. After Viewing & Kaiako Move (Process & Apply)
Kaiako Move: Use the memory rhyme for quotient rule: 'Low d-High minus High d-Low, over the square of Low we go!'.
Immediate Task: Add to your Level 3 Calculus revision logbook (section 3): Product & Quotient Rule Differentiation Sheet.
⚡ Whakaoho | Do Now: Two Functions Multiplied (10 mins)
Differentiate x³ and sin x separately from memory. Now predict: is the derivative of x³·sin x equal to 3x²·cos x? Test your prediction on x·x (which you already know differentiates to 2x). Hold on to what that test shows you.
📖 Activity 1: Product and Quotient Rules, and When Not to Use Them (25 mins)
Your Do Now proves the naive guess fails. Use the product rule (uv)′ = u′v + uv′ on x²sin x and eˣ ln x, then the quotient rule on (2x + 1)/(x − 3) and sin x / x. Then a judgement task: differentiate (x³ + 2x)/x. Do it once with the quotient rule and once by simplifying to x² + 2 first. Same answer, very different effort. Choosing the cheaper route is an assessed skill, not a shortcut.
📝 Activity 2: Exam-Style Practice — Rule Selection (20 mins)
For each of y = x⁴ eˣ, y = (3x − 1)/(x² + 2), y = x²·ln x: name the rule before differentiating, then apply it. For Merit, label u, v, u′ and v′ explicitly. For Excellence, explain why the quotient rule's numerator is u′v − uv′ and not u′v + uv′, and what would go wrong if the order were swapped.
🏫 Kaiako Planning & Pedagogy Notes
NCEA Level 3 Alignment: Direct preparation for NCEA Level 3 Calculus — External (Apply differentiation methods in solving problems). Emphasise complete algebraic working and clear context conclusions for Merit/Excellence grades.