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9.4PureStretch

The Product Rule

The product rule differentiates a product of two functions. You cannot just multiply the separate derivatives — each term keeps one factor and differentiates the other.

24 min Differentiation
Edexcel A level Maths: 9.4 and 9.5 The Product and Quotient RuleCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • State and apply the product rule
  • Differentiate products like x²eˣ
  • Combine with the chain rule
  • Simplify and factorise the result
1

The rule

If y=uvy = uv where uu and vv are functions of xx, then dydx=udvdx+vdudx\dfrac{dy}{dx} = u\dfrac{dv}{dx} + v\dfrac{du}{dx}, often written u′v+uv′u'v + uv'.

ddx(uv)=udvdx+vdudx\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}
Product rule.
Example 1Differentiate y=x2exy = x^2 e^x.
1u=x2u = x^2, v=exv = e^x.
2u′=2xu' = 2x, v′=exv' = e^x.
3dydx=x2ex+ex(2x)=ex(x2+2x)\tfrac{dy}{dx} = x^2 e^x + e^x(2x) = e^x(x^2 + 2x).
Answer
ex(x2+2x)e^x(x^2 + 2x)

Tip — Differentiate one, keep the other; add the two terms. Factor the answer where possible.

Formula recap

ddx(uv)=u′v+uv′\frac{d}{dx}(uv) = u'v + uv'
Product rule.

Common mistakes to avoid

Writing (uv)′ = u′v′.
It is u′v + uv′.
Forgetting the chain rule on a composite factor.
Differentiate each factor correctly, using the chain rule if needed.

Key takeaways

  • (uv)′ = u′v + uv′.
  • Differentiate one factor, keep the other, then add.
  • Factor the result to simplify.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The product rule gives (uv)′ =

    • u′v+uv′u'v + uv'
    • u′v′u'v'
    • u′v−uv′u'v - uv'
    • uv′uv'
    Show answer and explanation

    Answer: u′v+uv′u'v + uv'

    Sum of two terms.

  2. ddx(x2ex)\frac{d}{dx}(x^2 e^x) equals:

    • ex(x2+2x)e^x(x^2 + 2x)
    • 2xex2x e^x
    • x2exx^2 e^x
    • ex(x2−2x)e^x(x^2 - 2x)
    Show answer and explanation

    Answer: ex(x2+2x)e^x(x^2 + 2x)

    x²eˣ + 2x eˣ.

  3. ddx(xsin⁡x)\frac{d}{dx}(x\sin x) equals:

    • sin⁡x+xcos⁡x\sin x + x\cos x
    • cos⁡x\cos x
    • xcos⁡xx\cos x
    • sin⁡x−xcos⁡x\sin x - x\cos x
    Show answer and explanation

    Answer: sin⁡x+xcos⁡x\sin x + x\cos x

    u′v + uv′.

  4. Is (uv)′ = u′v′ correct?

    • No
    • Yes
    • Only if u = v
    • Only for exponentials
    Show answer and explanation

    Answer: No

    Common error.

  5. ddx(xe2x)\frac{d}{dx}(x e^{2x}) equals:

    • e2x(1+2x)e^{2x}(1 + 2x)
    • e2xe^{2x}
    • 2xe2x2x e^{2x}
    • e2x(2+x)e^{2x}(2 + x)
    Show answer and explanation

    Answer: e2x(1+2x)e^{2x}(1 + 2x)

    e^{2x} + x·2e^{2x}.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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