Lesson 4: The Chain Rule & Composite Function Differentiation
Applying differentiation methods, rules, optimisation, and related rates for NCEA Level 3 Calculus.
🎯 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
The Chain Rule — Khan Academy
Video (6 min 29 sec, Khan Academy): A comprehensive visual and worked-example introduction to the chain rule dy/dx = (dy/du)(du/dx), essential for differentiating composite functions. Covers the "inside-outside" strategy clearly.
🧠 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 6½-minute clip. In your calculus logbook, record:
- 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 (10 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 (25 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 (20 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.
🏫 Kaiako Planning & Pedagogy Notes
NCEA Level 3 Alignment: Direct preparation for Level 3 Calculus (Apply differentiation methods in solving problems). Emphasise complete algebraic working and clear context conclusions for Merit/Excellence grades.