Product and quotient rules: Class 11 practice
(uv)′ = u′v + uv′ and (u/v)′ = (u′v − uv′)/v². Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Power rule
The idea behind product and quotient rules
(uv)′ = u′v + uv′ and (u/v)′ = (u′v − uv′)/v². Identify u and v, differentiate each, then substitute into the rule.
Step-by-step method
- Name u and v.
- Find u′ and v′.
- Apply the rule and simplify.
- Substitute a value if asked.
Worked example 1 (Basic)
If y = (1x − 5)(2x² + 1), find dy/dx at x = 2.
- u′ = 1, v′ = 4x
- dy/dx = 1(2x² + 1) + (1x − 5)(4x)
- At x = 2: -15
- Answer: -15
Worked example 2 (Standard)
If y = (3x − 4)(4x² + 1), find dy/dx at x = -3.
- u′ = 3, v′ = 8x
- dy/dx = 3(4x² + 1) + (3x − 4)(8x)
- At x = -3: 423
- Answer: 423
Worked example 3 (Challenge)
If y = (4x − 5)/(x² + 1), find dy/dx at x = 2.
- (u′v − uv′)/v²
- = [4(x² + 1) − (4x − 5)(2x)]/(x² + 1)²
- At x = 2: 8/25
- Answer: 8/25
Common mistakes and misconceptions
- Writing (uv)′ = u′v′
- Reversing the order in the quotient rule numerator
Practice questions
- (Basic) If y = (4x − 1)(2x² + 1), find dy/dx at x = -1.
- (Basic) If y = (3x − 5)(3x² + 1), find dy/dx at x = 3.
- (Standard) If y = (2x − 4)(1x² + 1), find dy/dx at x = -3.
- (Standard) If y = (5x − 4)(3x² + 1), find dy/dx at x = -3.
- (Challenge) If y = (2x + 1)/(x² + 4), find dy/dx at x = -2.
- (Challenge) If y = (1x + 2)/(x² + 4), find dy/dx at x = -3.
Answer key with full working
- Q1: u′ = 4, v′ = 4x → dy/dx = 4(2x² + 1) + (4x − 1)(4x) → At x = -1: 32 → Answer: 32
- Q2: u′ = 3, v′ = 6x → dy/dx = 3(3x² + 1) + (3x − 5)(6x) → At x = 3: 156 → Answer: 156
- Q3: u′ = 2, v′ = 2x → dy/dx = 2(1x² + 1) + (2x − 4)(2x) → At x = -3: 80 → Answer: 80
- Q4: u′ = 5, v′ = 6x → dy/dx = 5(3x² + 1) + (5x − 4)(6x) → At x = -3: 482 → Answer: 482
- Q5: (u′v − uv′)/v² → = [2(x² + 4) − (2x + 1)(2x)]/(x² + 4)² → At x = -2: 1/16 → Answer: 1/16
- Q6: (u′v − uv′)/v² → = [1(x² + 4) − (1x + 2)(2x)]/(x² + 4)² → At x = -3: 7/169 → Answer: 7/169
Syllabus note
This skill is listed under "Limits and Derivatives" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is product and quotient rules in?
"Limits and Derivatives", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.