The quotient rule differentiates one function divided by another. If y = u/v, then dy/dx = (u′v − uv′)/v². The order of the subtraction matters: u′v comes first. Leave the denominator as v², and expand and simplify the numerator; stationary points come from setting the numerator equal to zero.
The quotient rule
- If y = u/v, then dy/dx = (u′v − uv′) / v²
- A fraction is zero only when its numerator is zero
- If the denominator is a single power of x, split the fraction instead
Memory aid: “low d-high minus high d-low, over low squared”.
What is the quotient rule?
When a function is one expression divided by another, such as x/(x² + 1), the quotient rule gives its derivative. Label u (the top) and v (the bottom), find u′ and v′, then use (u′v − uv′)/v².
Unlike the product rule, the order matters. Swapping the two terms in the numerator changes the sign of the answer.
Quadratic numerators and denominators
The rule is the same when u or v is a quadratic. Differentiate each carefully, expand the numerator fully and collect like terms. Leave the denominator as v²: expanding it gains nothing and makes cancelling impossible.
The quotient rule with the chain rule
If u or v is a composite, find its derivative with the chain rule first, then substitute. For example, u = (x + 1)² gives u′ = 2(x + 1). Look for a common factor in the numerator afterwards; factorising often leads to a tidy answer.
Stationary points of a quotient
A fraction equals zero only when its numerator is zero; the denominator can never make it zero. So to solve f′(x) = 0 for a quotient, set the numerator of f′(x) equal to zero and solve.
Quotient rule, or rewrite?
If the denominator is a single power of x, it is quicker to split the fraction and use the power rule. For example, (3x² + 1)/x = 3x + x⁻¹. Save the quotient rule for denominators that are genuine brackets or sums, such as x + 1 or x² + 1.
Quotients with eˣ, ln x and trig
The quotient rule works for any quotient, including those with a standard function on the top or bottom. Keep the order u′v − uv′ exactly. The derivative of tan x = sin x / cos x is itself a quotient rule result: (cos²x + sin²x)/cos²x = sec²x.
Worked examples
Example 1. Differentiate y = x/(x + 1).
- u = x, v = x + 1, u′ = 1, v′ = 1.
- ((1)(x + 1) − x(1)) / (x + 1)² = (x + 1 − x)/(x + 1)².
Answer: dy/dx = 1/(x + 1)².
Marks: M1 for the quotient rule, A1 for the answer.
Example 2. Differentiate y = x²/(x + 1), factorising the numerator.
- u = x², v = x + 1, u′ = 2x, v′ = 1.
- (2x(x + 1) − x²)/(x + 1)² = (x² + 2x)/(x + 1)².
Answer: dy/dx = x(x + 2)/(x + 1)².
Marks: M1 for the quotient rule, A1 for the factorised answer.
Example 3. Differentiate y = (x + 1)²/(x − 1), factorising the numerator.
- u = (x + 1)², v = x − 1, u′ = 2(x + 1), v′ = 1.
- Numerator: 2(x + 1)(x − 1) − (x + 1)² = (x + 1)[2(x − 1) − (x + 1)] = (x + 1)(x − 3).
Answer: dy/dx = (x + 1)(x − 3)/(x − 1)².
Marks: M1 for the chain rule on u, M1 for factorising, A1 for the answer.
Example 4. Find the x-coordinate of the stationary point of y = x/(x² + 4) for x ≥ 0.
- dy/dx = ((x² + 4) − x(2x))/(x² + 4)² = (4 − x²)/(x² + 4)².
- Set the numerator to zero: 4 − x² = 0, so x = ±2.
- Only x = 2 satisfies x ≥ 0.
Answer: x = 2.
Marks: M1 for the numerator 4 − x², A1 for x = 2.
Example 5. Differentiate y = (sin x)/x.
- u = sin x, v = x, u′ = cos x, v′ = 1.
- (x cos x − sin x)/x².
Answer: dy/dx = (x cos x − sin x)/x².
Marks: M1 for the quotient rule, A1 for the answer.
Common mistakes
- Subtracting in the wrong order. The numerator is u′v − uv′. Reversing it changes the sign.
- Squaring the wrong function. The denominator is v², the bottom of the original fraction, squared.
- Expanding v². Leave (x + 1)² as it is; you may need to cancel a factor.
- Setting the denominator to zero. Stationary points come from the numerator of dy/dx.
- Using the quotient rule when splitting is easier. (3x² + 1)/x is simply 3x + x⁻¹.
How the quotient rule is examined
Quotient rule questions sit within Pure 3 (WMA13) for Edexcel International A-Level, and within Year 2 Pure for Edexcel A-Level (9MA0). They usually ask you to:
- show that a derivative simplifies to a given fraction
- find the stationary points of a rational function
- find a tangent or normal at a given point
- prove a derivative such as tan x giving sec²x
Full quotient rule questions, with mark schemes, appear in the Pure 3 differentiation exam questions. Use the differentiation worksheet or the differentiation revision engine for more.
Strong algebraic fraction skills make quotient rule answers much easier to simplify. It works alongside the product rule and the standard derivatives.
Frequently asked questions
What is the quotient rule?
If y = u/v, then dy/dx = (u′v − uv′)/v².
Can I use the product rule instead of the quotient rule?
Yes, by writing u/v as u × v⁻¹ and using the chain rule on v⁻¹. It gives the same answer, but the quotient rule is usually quicker.
How do you find stationary points with the quotient rule?
Differentiate, then set the numerator of the derivative equal to zero. The denominator never makes a fraction zero.