All worked examples

The product rule

Differentiation
A-level Maths (9MA0)
ORIGINAL

Differentiate a product of two functions — then factorise, because the mark scheme wants it tidy.

Verified against Edexcel 9MA0 (2026 spec)

Question

Differentiate y=x2exy = x^2 e^x with respect to xx.

Every step worked, with the reasoning.

  1. 1
    Let u=x2u = x^2 and v=exv = e^x.

    It's a product of two functions of xx.

  2. 2
    u′=2xu' = 2x and v′=exv' = e^x

    Differentiate each factor.

  3. 3
    dydx=u′v+uv′=2x ex+x2ex\dfrac{dy}{dx} = u'v + uv' = 2x\,e^x + x^2 e^x

    Product rule: u′v+uv′u'v + uv'.

  4. 4
    dydx=xex(x+2)\dfrac{dy}{dx} = x e^x(x + 2)

    Factor out xexx e^x.

Answer: dydx=xex(x+2)\dfrac{dy}{dx} = x e^x(x + 2)

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