Differentiate quotients of functions
Apply the quotient rule with a clearly identified numerator and denominator, then simplify and retain domain restrictions.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
For y = u/v, which expression is the quotient-rule numerator?
What you will learn
- Use d(u/v)/dx = (u′v − uv′)/v²
- Keep the quotient-rule order and denominator square correct
- Combine the quotient rule with the chain rule where needed
The numerator keeps a specific order
For y = u/v with v ≠ 0, y′ = (u′v − uv′)/v². The first product is derivative-of-top times bottom; subtract top times derivative-of-bottom; divide by bottom squared.
The order matters. Reversing the two numerator terms changes the sign. A quick label above the fraction helps distinguish top u from bottom v.
The original denominator still limits the domain
For y = (x² + 1)/(x − 1), the input x = 1 is excluded before and after differentiation. Simplifying the derivative does not make the original function defined there.
You may sometimes simplify the function algebraically before differentiating, but keep any original hole or excluded input in view.
Pause and explain
If y = x/(x + 1), what is the squared denominator in y′?
Worked example
Differentiate y = (x² + 1)/(x − 1).
Show the worked solution
- Let u = x² + 1 and v = x − 1. Then u′ = 2x and v′ = 1.
- The numerator is 2x(x − 1) − (x² + 1)(1) = x² − 2x − 1.
- Divide by (x − 1)²; x = 1 remains excluded.
Answer dy/dx = (x² − 2x − 1)/(x − 1)², for x ≠ 1.
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Differentiate y = (x + 2)/(x² + 1).
Give me a hint
Identify top u = x + 2 and bottom v = x² + 1.
Compare my reasoning
- u′ = 1 and v′ = 2x.
- The numerator is (x² + 1) − 2x(x + 2).
- Simplify and divide by (x² + 1)².
dy/dx = (−x² − 4x + 1)/(x² + 1)².
Look for these in your work
- Top and bottom identified
- Numerator order correct
- Denominator squared
Differentiate y = sin x/x for x ≠ 0.
Give me a hint
Apply the quotient rule; the derivative of the denominator is 1.
Compare my reasoning
- u = sin x and v = x, so u′ = cos x and v′ = 1.
- The numerator is x cos x − sin x.
- Divide by x².
dy/dx = (x cos x − sin x)/x², for x ≠ 0.
Look for these in your work
- Trig derivative uses radians
- Quotient order correct
- Excluded input retained
Differentiate f(x) = (3x² + 1)/(2x − 5). State the excluded input.
Give me a hint
The denominator derivative is 2; keep the quotient-rule subtraction in order.
Compare my reasoning
- u′ = 6x and v′ = 2.
- Numerator = 6x(2x − 5) − 2(3x² + 1) = 6x² − 30x − 2.
- Denominator is (2x − 5)² and x ≠ 5/2.
f′(x) = (6x² − 30x − 2)/(2x − 5)², for x ≠ 5/2.
Look for these in your work
- Both derivatives found
- Numerator simplified
- Original denominator restriction stated
Common mistakes
- Using uv′ − u′v instead of u′v − uv′
- Forgetting to square the denominator
- Differentiating a quotient as a quotient of derivatives
- Dropping an excluded denominator value after simplification
What can you explain now?
What is the denominator in the quotient-rule derivative of u/v?
Compare with the explanation
v²
The rule divides by the square of the original denominator.
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Make it stick
What is the order of the terms in the quotient rule, and why is the denominator squared?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.