Study guides › Level 10 · Calculus

The chain rule

Level 10 · Calculus · for questions like “f(x) = (2x + 1)³. What is f′(1)?”

💡 The idea

For a function inside a function: derivative of the outside (leave the inside alone) × derivative of the inside.

d/dx (ax + b)ⁿ = n(ax + b)ⁿ⁻¹ × a

✏️ Worked examples

1. f(x) = (2x + 1)³: f′(x) = 3(2x + 1)² × 2. At x = 1 the bracket is 3, so 3 × 9 × 2 = 54.

2. f(x) = (3x + 2)²: f′(x) = 2(3x + 2) × 3. At x = 0 that's 2 × 2 × 3 = 12.

⚠️ Watch out

• Don't forget × the inside's derivative (the a). Leaving it out is the classic mistake.

🎯 Try these

1. (x + 4)², f′(1)
2. (2x + 3)², f′(0)
3. (4x + 1)³, f′(0)

Answers: 1) 10 2) 12 3) 12

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