Lesson 3: The Product Rule & Quotient Rule
Two functions multiplied is not two derivatives multiplied. The product and quotient rules — and the judgement to notice when neither of them is needed.
🎯 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 2 min 39 sec clip on the product rule. In your calculus logbook, record:
- The product rule formula: Write d/dx [u·v] = u′v + uv′, then copy the video's worked example line by line, labelling which factor is u and which is v.
- Why the naive guess fails: In your own words, why is d/dx [u·v] not u′v′? Use your Do Now test on x·x as the counter-example.
- Application prep: Differentiate f(x) = x³ sin x using the product rule. (Answer: f′(x) = 3x² sin x + x³ cos x.)
🗣️ 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 (5 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 (20 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 (15 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.
🎫 Exit Ticket: Execute, Then Choose (5 mins)
(a) Differentiate y = x² cos x. (b) Differentiate y = (2x + 1)/(x − 3). (c) In one sentence: give a function where using the quotient rule would be a waste of time, and say why.
🏫 Kaiako Planning & Pedagogy Notes
NCEA Level 3 alignment: Direct preparation for NCEA Level 3 Achievement Standard AS 91578, Apply differentiation methods in solving problems (external, 6 credits), against The New Zealand Curriculum (2007) Mathematics and Statistics Level 8 — Calculus. Emphasise complete algebraic working and a conclusion written in the context of the question, which is what separates Merit from Achieved.
Materials: Mini-whiteboards, graphics calculator, calculus logbook. Pre-rule a u / v / u′ / v′ grid on the board; ākonga who label before they differentiate make far fewer sign errors.
Pacing (53 mins): Do Now 5 · media anchor 8 (the clip plus the before- and after-viewing prompts) · Activity 1 20 · Activity 2 15 · exit ticket 5.
Formative assessment — what to look for: Parts (a) and (b) are execution and most of the class will get them. Part (c) is rule selection, and it is the Merit-to-Excellence separator in this topic: an ākonga who reaches for the quotient rule on (x³ + 2x)/x has not understood what the rule is for.
Differentiation. Entry: Give the u and v split, and ask only for u′, v′ and the substitution. On level: The full set, choosing the split independently. Extension: Derive the quotient rule from the product rule: write u/v as u · v⁻¹, differentiate with the product and chain rules, and simplify until you reach (u′v − uv′)/v².