Quotient Rule

This lesson covers the quotient rule for differentiating one function divided by another, as required by the Edexcel A-Level Mathematics specification (9MA0). You also need to understand the connection between the quotient rule and the product rule.

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When to Use the Quotient Rule

Use the quotient rule when you need to differentiate y = u/v, where u and v are both functions of x and you cannot simplify the fraction into separate terms.

Examples where the quotient rule is needed:

• y = sin x / x
• y = eˣ / (x + 1)
• y = x² / (2x - 3)
• y = ln x / x

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The Quotient Rule Formula

If y = u/v, where u = u(x) and v = v(x), then:

dy/dx = (v(du/dx) - u(dv/dx)) / v²

Or in shorter notation:

dy/dx = (vu' - uv') / v²

In words: "(bottom × derivative of top - top × derivative of bottom) / bottom squared."

Exam Tip: The quotient rule is given in the formula booklet at A-Level. However, you should practise it enough that you can apply it quickly and accurately.

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Step-by-Step Method