Product Rule

This lesson covers the product rule for differentiating the product of two functions, as required by the Edexcel A-Level Mathematics specification (9MA0). You need to know when to use the product rule and how to apply it correctly.

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When to Use the Product Rule

Use the product rule when you need to differentiate y = u × v, where u and v are both functions of x, and it is not practical to expand the expression first.

Examples where the product rule is needed:

• y = x² sin x (polynomial × trig)
• y = xeˣ (polynomial × exponential)
• y = (2x + 1) ln x (linear × logarithmic)
• y = eˣ cos x (exponential × trig)

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The Product Rule Formula

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

dy/dx = u(dv/dx) + v(du/dx)

Or in shorter notation:

dy/dx = uv' + vu'

In words: "(first × derivative of second) + (second × derivative of first)."

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Step-by-Step Method