The NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Miscellaneous Exercise solve every mixed question of the final exercise, according to the latest 2026-27 CBSE syllabus. These solutions help students prepare for CBSE Boards, JEE Main, JEE Advanced and CUET.
Each solution in this Collegedunia set is prepared by subject experts, based on the 2026-27 NCERT textbook, and checked against recent CBSE exam patterns.
The Miscellaneous Exercise blends the limit and derivative ideas from both earlier exercises into tougher, exam-style problems. The exercise has 30 questions built around these skills:
- Differentiating harder polynomial, rational and trigonometric functions.
- Applying the product and quotient rules to combined expressions.
- Using first principle and standard limits on mixed problems.
Topics Covered in Miscellaneous Exercise
This exercise revisits every tool from Chapter 12 and pushes it further. Students meet functions that need more than one rule, along with trigonometric derivatives and limits that combine several standard results. The Miscellaneous Exercise is the closest match to the level asked in board and entrance exams.
- Advanced derivatives: differentiating products and quotients of polynomials and trigonometric functions.
- Trigonometric derivatives: tan x, sec x, cosec x and combinations built from sin x and cos x.
- First principle on mixed functions: deriving results from the limit definition for harder cases.
- Standard limits: combining sin x / x with algebraic limits in one problem.
- Rearranging before differentiating: simplifying an expression so a rule applies cleanly.
Important: many questions become far shorter after a small simplification, so it pays to tidy the expression before choosing a rule.
Question-wise Breakdown of Miscellaneous Exercise
The NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Miscellaneous Exercise solve all 30 questions in order. The table below groups the questions by type so students can plan their practice. Full step-by-step answers are inside the PDF above.
| Questions | What they ask | Skill tested |
|---|---|---|
| Q1 to Q8 | Derivatives from first principle for mixed functions | First principle |
| Q9 to Q18 | Derivatives of polynomials using power and sum rules | Algebra of derivatives |
| Q19 to Q25 | Product and quotient rule problems | Product and quotient rules |
| Q26 to Q30 | Trigonometric and combined function derivatives | Trigonometric derivatives |
Because the questions mix every method, the Miscellaneous Exercise is the best final check before a chapter test.
Key Formulas and Definitions for Miscellaneous Exercise
This exercise uses the full toolkit of Chapter 12. Students should have all of the results below ready before starting.
| Rule | Result |
|---|---|
| First principle | f′(x) = limh→0 [f(x + h) − f(x)] / h |
| Power rule | d/dx(xn) = n·xn−1 |
| Product rule | (u·v)′ = u′v + uv′ |
| Quotient rule | (u/v)′ = (u′v − uv′) / v2 |
| Trigonometric | d/dx(sin x) = cos x, d/dx(cos x) = −sin x |
| Standard limit | limx→0 (sin x) / x = 1 |
| Algebraic limit | limx→a (xn − an) / (x − a) = n·an−1 |
Tip: for a ratio such as (sin x) / x² near 0, split it as (sin x / x) · (1 / x) so the standard limit can be used.
Common Mistakes in Miscellaneous Exercise
Students lose marks in the Miscellaneous Exercise on method and simplification errors. The list below covers the slips seen most often in tests on the NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Miscellaneous Exercise.
- Choosing the wrong rule because the expression was not simplified first.
- Applying the quotient rule with the terms in the wrong order, giving a sign error.
- Forgetting to differentiate every factor in a product of three terms.
- Mixing up the derivative of a trigonometric function with its limit value.
Practice Questions for Miscellaneous Exercise
After reading the solutions, students should test themselves on the same mixed problems. The card below opens a set of solved practice questions with step-by-step answers for the Miscellaneous Exercise.
Practice Card: Solved Practice Questions for Class 11 Maths Limits and Derivatives Miscellaneous Exercise — attempt each question, then check the worked solution.
Other Exercises of Chapter 12 Limits and Derivatives
Chapter 12 Limits and Derivatives has two exercises plus this Miscellaneous Exercise. Use the table to revisit an earlier exercise if a method needs revision.
| Exercise | NCERT Solutions link |
|---|---|
| Exercise 12.1 | Class 11 Maths Limits and Derivatives Exercise 12.1 Solutions |
| Exercise 12.2 | Class 11 Maths Limits and Derivatives Exercise 12.2 Solutions |
More Class 11 Maths Limits and Derivatives Resources
Pair the solutions with the notes and the NCERT book PDF for full chapter revision. All three resources for Chapter 12 Limits and Derivatives are linked below.
| Resource | Link |
|---|---|
| Revision Notes | Class 11 Maths Limits and Derivatives Notes |
| Handwritten Notes | Class 11 Maths Limits and Derivatives Handwritten Notes |
| NCERT Book PDF | Class 11 Maths Limits and Derivatives NCERT Book PDF |
NCERT Solutions for Other Class 11 Maths Chapters
Students can move to the first exercise of any other Class 11 Maths chapter using the cross-sell table below.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1: Sets | Sets Solutions |
| Chapter 2: Relations and Functions | Relations and Functions Solutions |
| Chapter 3: Trigonometric Functions | Trigonometric Functions Solutions |
| Chapter 4: Complex Numbers and Quadratic Equations | Complex Numbers Solutions |
| Chapter 5: Linear Inequalities | Linear Inequalities Solutions |
| Chapter 6: Permutations and Combinations | Permutations and Combinations Solutions |
| Chapter 7: Binomial Theorem | Binomial Theorem Solutions |
| Chapter 8: Sequences and Series | Sequences and Series Solutions |
| Chapter 9: Straight Lines | Straight Lines Solutions |
| Chapter 10: Conic Sections | Conic Sections Solutions |
| Chapter 11: Introduction to Three Dimensional Geometry | Three Dimensional Geometry Solutions |
| Chapter 12: Limits and Derivatives | You are here |
| Chapter 13: Statistics | Statistics Solutions |
| Chapter 14: Probability | Probability Solutions |
Student Feedback
In a Collegedunia survey of 10,210 Class 11 students conducted before the 2026 exams, 69% of students said the product and quotient rule questions in the Miscellaneous Exercise were the most useful for board practice, while first-principle problems built the clearest understanding.
FAQs on NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Miscellaneous Exercise
Limits and Derivatives NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Class 11 Maths Chapter 12 Limits and Derivatives Miscellaneous Exercise?
Ans. The Miscellaneous Exercise has 30 questions. They mix derivatives from first principle, the power rule, the product and quotient rules, and trigonometric derivatives, along with a few combined limit problems.
Ques. Is the Miscellaneous Exercise harder than Exercise 12.1 and 12.2?
Ans. The Miscellaneous Exercise is more demanding because each question can need more than one rule and often a simplification first. It is the closest match to the level of questions asked in CBSE Boards, JEE Main and CUET.
Ques. Which rules are used most in the Miscellaneous Exercise?
Ans. The product rule (u·v)′ = u′v + uv′ and the quotient rule (u/v)′ = (u′v − uv′) / v2 are used most, alongside the power rule and the trigonometric derivatives. Several questions combine two of these rules in one problem.
Ques. How should students prepare for the Miscellaneous Exercise?
Ans. Students should be confident with limits from Exercise 12.1 and derivatives from Exercise 12.2 before starting. Then they should practise simplifying each expression, choosing the right rule and checking signs, since these steps decide most marks in the Miscellaneous Exercise.
Ques. Are the NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Miscellaneous Exercise free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Miscellaneous Exercise are free to download as a PDF from this page. Every question is solved step by step, according to the 2026-27 NCERT textbook, so students can revise offline for CBSE Boards, JEE Main, JEE Advanced and CUET.
Comments