The Chain Rule: How to Differentiate Composite Functions
Learn the chain rule in three steps with two worked examples and the most common mistake.
The rule
- d/dx [f(g(x))] = f′(g(x)) · g′(x)
Differentiate the outer function, keep the inside, then multiply by the derivative of the inside.
Example 1
Differentiate y = (3x + 1)⁵
- Outer: u⁵ → 5u⁴
- Inside: 3x + 1 → 3
- 5(3x + 1)⁴ · 3
15(3x + 1)⁴
Example 2
Differentiate y = sin(x²)
- Outer: sin u → cos u
- Inside: x² → 2x
- cos(x²) · 2x
2x cos(x²)
Common mistake
d/dx (3x + 1)⁵ = 5(3x + 1)⁴
Do not forget to multiply by the derivative of the inside.
Leave the inside function unchanged when you differentiate the outside.
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