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9.5PureStretch

The Quotient Rule

The quotient rule differentiates a fraction of two functions. The order of the numerator terms matters — get the minus sign in the right place.

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 quotient rule
  • Differentiate fractions of functions
  • Get the sign and order right
  • Derive tan x as an example
1

The rule

If y=uvy = \dfrac{u}{v}, then dydx=v u′−u v′v2\dfrac{dy}{dx} = \dfrac{v\,u' - u\,v'}{v^2}. The numerator is “bottom times derivative of top minus top times derivative of bottom”.

ddx(uv)=vu′−uv′v2\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v u' - u v'}{v^2}
Quotient rule.
Example 1Differentiate y=xx+1y = \dfrac{x}{x+1}.
1u=xu = x, v=x+1v = x+1, u′=1u' = 1, v′=1v' = 1.
2(x+1)(1)−x(1)(x+1)2=1(x+1)2\tfrac{(x+1)(1) - x(1)}{(x+1)^2} = \tfrac{1}{(x+1)^2}.
Answer
1(x+1)2\dfrac{1}{(x+1)^2}

Tip — Mnemonic: “low d-high minus high d-low, over low squared.”

2

Deriving tan x

Writing tan⁡x=sin⁡xcos⁡x\tan x = \dfrac{\sin x}{\cos x} and applying the quotient rule gives ddx(tan⁡x)=sec⁡2x\dfrac{d}{dx}(\tan x) = \sec^2 x.

Formula recap

ddx ⁣(uv)=vu′−uv′v2\frac{d}{dx}\!\left(\frac{u}{v}\right) = \frac{v u' - u v'}{v^2}
Quotient rule.
ddx(tan⁡x)=sec⁡2x\frac{d}{dx}(\tan x) = \sec^2 x
Derived from it.

Common mistakes to avoid

Writing the numerator as u′v − uv′ (wrong order).
It is v u′ − u v′ (bottom times d-top first).
Forgetting to square the denominator.
Divide by v².

Key takeaways

  • (u/v)′ = (v u′ − u v′)/v².
  • “Low d-high minus high d-low over low squared.”
  • d/dx(tan x) = sec²x.

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 quotient rule numerator is:

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

    Answer: vu′−uv′v u' - u v'

    Bottom × d-top first.

  2. The quotient rule denominator is:

    • v2v^2
    • vv
    • u2u^2
    • uvuv
    Show answer and explanation

    Answer: v2v^2

    Square the bottom.

  3. ddxxx+1\frac{d}{dx}\frac{x}{x+1} equals:

    • 1(x+1)2\frac{1}{(x+1)^2}
    • 1x+1\frac{1}{x+1}
    • x(x+1)2\frac{x}{(x+1)^2}
    • 11
    Show answer and explanation

    Answer: 1(x+1)2\frac{1}{(x+1)^2}

    Apply the rule.

  4. ddx(tan⁡x)\frac{d}{dx}(\tan x) equals:

    • sec⁡2x\sec^2 x
    • sec⁡xtan⁡x\sec x\tan x
    • −csc⁡2x-\csc^2 x
    • cos⁡2x\cos^2 x
    Show answer and explanation

    Answer: sec⁡2x\sec^2 x

    From sin/cos quotient.

  5. A common error is to:

    • swap the numerator order
    • square the top
    • add the derivatives
    • use radians
    Show answer and explanation

    Answer: swap the numerator order

    Order matters for the minus.

Next step

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