Chain Rule

This lesson covers the chain rule for differentiating composite functions as required by the Edexcel A-Level Mathematics specification (9MA0). The chain rule is one of the most important differentiation techniques and is used extensively throughout A-Level Mathematics.

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What Is a Composite Function?

A composite function is a "function of a function" — one function is applied inside another.

Examples:

• (3x + 1)⁵ — a power of a linear expression
• sin(2x) — sin applied to 2x
• e^(x²) — exponential of x²
• √(4x - 1) — square root of a linear expression
• ln(x² + 5) — ln of a quadratic

These cannot be differentiated using the power rule alone — they require the chain rule.

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The Chain Rule

If y = f(g(x)) — that is, y is a function of u, where u is a function of x — then:

dy/dx = (dy/du) × (du/dx)

Or equivalently, using function notation:

d/dx[f(g(x))] = f'(g(x)) × g'(x)

In words: "differentiate the outer function (keeping the inner function unchanged), then multiply by the derivative of the inner function."

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The Chain Rule for Powers