Lesson 4: The Chain Rule & Composite Function Differentiation
Something inside something else. Differentiate the outside, leave the inside alone, then multiply by the inside’s derivative — and never, under any pressure, lose that last factor.
🎯 Ngā Whāinga Akoranga | Learning Intentions
The Chain Rule dy/dx = (dy/du) * (du/dx) for composite nested functions y = f(g(x)).
Identify inner g(x) and outer f(u) functions, differentiate complex nested algebraic, trig, and log expressions, and combine with product/quotient rules.
🎥 Media Anchor & Pedagogical Scaffold
Common Chain Rule Misunderstandings — Khan Academy
Video (6 min 48 sec, Khan Academy): Not an introduction — a misconceptions clinic. It works through the errors students most often make when applying the chain rule to composite functions. Teach the rule first (Do Now and Activity 1 below), then use this clip to stress-test it.
🧠 1. Before Viewing (Activate & Predict)
How do you differentiate a complex nested expression like y = (3x^2 + 5x)^7 or y = sin(e^(2x))?
👁️ 2. During Viewing (Watch With a Job)
Watch the full 6 min 48 sec clip. In your calculus logbook, record:
- Every misunderstanding the video names: Write each one down, and beside it the correct move. These are the marks you are most likely to lose in the external.
- The chain rule formula: Write dy/dx = (dy/du)(du/dx) and draw the "flow diagram" showing how the derivatives cascade.
- An example: For y = (3x + 2)⁵, identify u = 3x + 2 (inner), y = u⁵ (outer), and compute dy/dx = 5u⁴ × 3 = 15(3x + 2)⁴.
- Memory aid: Write down the phrase 'differentiate the outside, leave the inside alone, multiply by derivative of inside' in your own words.
🗣️ 3. After Viewing & Kaiako Move (Process & Apply)
Kaiako Move: Emphasise 'differentiate the outside, leave the inside alone, then multiply by derivative of the inside'.
Immediate Task: Add to your Level 3 Calculus revision logbook (section 4): Chain Rule & Composite Function Master Sheet.
⚡ Whakaoho | Do Now: Spotting the Inside Function (5 mins)
For each of (3x + 1)⁵, sin(x²) and e^(2x), write down what the inside function is and what is being done to it. Do not differentiate yet. Then differentiate (3x + 1)² by expanding it fully first, and keep that answer for the next activity.
📖 Activity 1: The Chain Rule, Checked Against Expansion (20 mins)
Apply dy/dx = dy/du · du/dx to (3x + 1)⁵, sin(x²) and e^(2x). Then verify: use the chain rule on (3x + 1)² and check it matches the expanded answer from your Do Now. That check is the point — the chain rule has to agree with a method you already trust. Finish with two that need the chain rule INSIDE the product rule: y = x²·e^(3x) and y = x·sin(2x).
📝 Activity 2: Exam-Style Practice — Layered Functions (15 mins)
Differentiate y = (x² + 4)⁷, y = ln(5x − 2) and y = e^(sin x). For Merit, state u and du/dx before combining. For Excellence, differentiate y = sin(3x²) and explain in a sentence which layer each factor of your answer came from.
🎫 Exit Ticket: The Inner Derivative (5 mins)
(a) Differentiate y = (3x² + 1)⁵. (b) Differentiate y = e²ₓ. (c) A student writes d/dx[sin(3x)] = cos(3x). Name the error in one sentence.
🏫 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. Keep the Lesson 3 expansion check available: (3x + 2)² can be expanded and differentiated directly, which is how ākonga verify the chain rule rather than trusting it.
Pacing (57 mins): Do Now 5 · media anchor 12 (the clip plus the before- and after-viewing prompts) · Activity 1 20 · Activity 2 15 · exit ticket 5.
Formative assessment — what to look for: Part (c) is the single most common lost mark in the AS 91578 external. If more than about a third of the class cannot name the missing inner derivative, the chain rule is not secure — do not move on to Lesson 5, because tangents, optimisation and related rates all sit on top of it.
Differentiation. Entry: Write u = (inside) and y = (outside) as two separate lines before differentiating anything. Two lines, every time, no exceptions. On level: The layered set, including trigonometric and exponential outsides. Extension: Differentiate y = sin(e²ₓ), a double composite, and then explain why the chain can be extended to any number of layers without a new rule.