Chain Rule

📖 What is the Chain Rule?

The chain rule is used to differentiate composite functions — functions inside other functions. If y depends on u, and u depends on x, then the chain rule links all three derivatives together.

d/dx[f(g(x))] = f'(g(x)) · g'(x)

📝 Step-by-Step Examples

1

Differentiate y = (3x + 2)⁵

Outer: u⁵ → 5u⁴ Inner: 3x+2 → 3

dy/dx = 5(3x+2)⁴ × 3 = 15(3x+2)⁴

2

Differentiate y = sin(x²)

Outer: sin(u) → cos(u) Inner: x² → 2x

dy/dx = cos(x²) × 2x = 2x·cos(x²)

3

Differentiate y = √(x³ + 1)

Write as (x³+1)^(1/2), outer → ½u^(-1/2), inner → 3x²

dy/dx = (3x²) / (2√(x³+1))

💡 How to Identify Composite Functions

Ask: "Is there a function inside another function?" If yes, use the chain rule.

sin(3x) — sin of something → chain rule needed

(x²+1)⁶ — power of something → chain rule needed

eˣ² — e to the power of something → chain rule needed

📖 What is the Chain Rule?

📝 Step-by-Step Examples

💡 How to Identify Composite Functions

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