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 Video by Zeeshan Zamurred Differentiation

Edexcel A level Maths: 9.4 and 9.5 The Product and Quotient RuleWatch the full walkthrough 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

The rule

If $y = \frac{u}{v}$ , then $\frac{d y}{d x} = \frac{v u ^{'} - u v ^{'}}{v ^{2}}$ . The numerator is “bottom times derivative of top minus top times derivative of bottom”.

\frac{d}{d x} (\frac{u}{v}) = \frac{v u ^{'} - u v ^{'}}{v ^{2}}

Quotient rule.

u = x

v = x + 1

u^{'} = 1

v^{'} = 1

\frac{( x + 1 ) ( 1 ) - x ( 1 )}{( x + 1 ) ^{2}} = \frac{1}{( x + 1 ) ^{2}}

Answer

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

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

Deriving tan x

Writing $tan x = \frac{sin x}{cos x}$ and applying the quotient rule gives $\frac{d}{d x} (tan x) = sec^{2} x$ .

Formula recap

\frac{d}{d x} (\frac{u}{v}) = \frac{v u ^{'} - u v ^{'}}{v ^{2}}

Quotient rule.

\frac{d}{d x} (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.

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