NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exercise 5.3 cover all 15 questions on implicit differentiation and on derivatives of inverse trigonometric functions. Collegedunia's subject experts solve each question with clear, exam-ready steps that match the 2026-27 NCERT print. The free PDF of these solutions is available to download on this page.
These solutions are written by experienced Class 12 Maths teachers and checked against the official NCERT answer key. Every question shows the formula used, the differentiation, and the final boxed answer on separate lines.
- CBSE Weightage: 8-10 marks (Continuity and Differentiability plus Applications of Derivatives, Calculus unit total 35 marks)
- JEE Main Weightage: 6-8% of the Mathematics section, with 2-3 questions on implicit and inverse-trig differentiation
- Exercise 5.3 Question Count: 15 questions covering implicit dy/dx extraction and direct differentiation of inverse trig identities
Exercise 5.3 at a glance: 15 questions, 7 of which need algebraic isolation of dy/dx after differentiating both sides. Implicit differentiation has featured in at least one CBSE board question every year since 2018.
Each solution lays out the implicit-differentiation pass, the term-by-term differentiation, and the final isolation of dy/dx on a separate line, so you can copy the same structure on your CBSE board answer script.
Continuity and Differentiability Ex 5.3 Solved Step by Step (Video)
Source: NCERT Wallah on YouTube
How will Collegedunia's NCERT Solutions for Exercise 5.3 Help You?
Exercise 5.3 is graded hard because students often forget the chain rule on every y-term, or make a sign error while isolating dy/dx. Collegedunia's solution PDF fixes exactly this.
- Every differentiation pass shown on a fresh line, with a clear note of where the chain rule was used, so the steps are easy to check.
- Formula recall printed at the top of any question that needs a less common inverse trig identity.
- Expert Solution block showing the alternate route via explicit y = f(x), where possible, so you can cross-check your answer.
- Tip callouts at common mistake points: forgetting dy/dx on y-terms, sign errors on inverse cotangent and cosecant, and mis-applying the quotient rule.
Important Concepts Covered in Class 12 Maths Exercise 5.3
Exercise 5.3 stays inside the four concept boxes below. Reviewing them before attempting the questions cuts solving time by roughly half.
| Concept | Working Definition | Used In Q No. |
|---|---|---|
| Implicit differentiation | Differentiate both sides of F(x, y) = 0 treating y = y(x) ; isolate dydx | 1, 2, 3, 4, 5 |
| Inverse sine and cosine derivatives | ddxsin-1x = 1√1-x2, ddxcos-1x = -1√1-x2 | 6, 7, 8, 9 |
| Inverse tangent and cotangent derivatives | ddxtan-1x = 11+x2, ddxcot-1x = -11+x2 | 10, 11, 12 |
| Inverse secant and cosecant derivatives | ddxsec-1x = 1|x|√x2-1 | 13, 14, 15 |
The inverse-secant and inverse-cosecant questions at the end of Ex 5.3 are the most-missed on CBSE boards because students drop the modulus on x. The Expert Solution explains why the modulus must stay, using the domain restriction of sec-1.
Exam Relevance of Continuity and Differentiability Exercise 5.3
Continuity and Differentiability is part of the Calculus unit, which carries 35 marks in the CBSE Class 12 Maths paper. Exercise 5.3 contributes most of the 4-mark implicit-differentiation questions. A quick look at recent board and JEE appearances:
| Exam Year | Question Type from Ex 5.3 | Marks |
|---|---|---|
| CBSE Board 2025 | Find dydx if x2 + xy + y2 = 100 | 4 |
| CBSE Board 2024 | Differentiate sin-1(2x√1-x2) with respect to x | 4 |
| CBSE Sample Paper 2024 | Implicit differentiation of sin(xy) + cos y = x | 3 |
| JEE Main 2025 (Jan session) | Inverse-tangent composite differentiation | 4 |
| JEE Main 2026 | Pending (exam rescheduled) | - |
Implicit differentiation is also the base skill for tangent-and-normal problems in Chapter 6, so time spent on Ex 5.3 pays off directly in Application of Derivatives.
Common Mistakes Students Make in Exercise 5.3
These solutions follow the same formal notation as the official NCERT print, line by line. Three error patterns cause most of the marks lost on this exercise in CBSE board scripts.
- Forgetting to multiply by dy/dx whenever a y-term is differentiated. This single slip costs 1-2 marks on an otherwise correct answer.
- Algebraic isolation errors after differentiation. The two-column isolation layout in the solutions PDF prevents this.
- Dropping the absolute value in the derivative of sec-1x, which CBSE marks as a complete error.
Other Resources for Class 12 Maths Chapter 5 Continuity and Differentiability
Pair the Exercise 5.3 solutions with the rest of the Chapter 5 resource library for complete preparation.
| Resource | What it covers |
|---|---|
| Chapter 5 Full Solutions | Every exercise of the chapter, solved in one place |
| Chapter 5 Notes | Theory, definitions, solved examples |
| Chapter 5 Formula Sheet | All identities on one page |
| Chapter 5 Handwritten Notes | Scanned-style revision notes |
Exercise-wise Breakdown of the Continuity and Differentiability Chapter
The Continuity and Differentiability chapter splits into 7 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the concept it tests.
| Exercise | Topic Tested |
|---|---|
| Exercise 5.1 | Continuity at a point and on an interval |
| Exercise 5.2 | Algebra of continuous functions |
| Exercise 5.3 | Differentiability and chain rule |
| Exercise 5.4 | Derivatives of inverse trigonometric functions |
| Exercise 5.5 | Logarithmic differentiation |
| Exercise 5.6 | Parametric and implicit differentiation |
| Exercise 5.7 | Second-order derivatives; Rolle's and Mean Value Theorem |
| Miscellaneous Exercise | Mixed continuity and differentiability problems |
NCERT Solutions for Class 12 Maths: All Chapters
Chapter-by-chapter NCERT Solutions for the rest of Class 12 Maths, aligned to the 2026-27 print.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1 | Relations and Functions NCERT Solutions |
| Chapter 2 | Inverse Trigonometric Functions NCERT Solutions |
| Chapter 3 | Matrices NCERT Solutions |
| Chapter 4 | Determinants NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 7 | Integrals NCERT Solutions |
| Chapter 8 | Application of Integrals NCERT Solutions |
| Chapter 9 | Differential Equations NCERT Solutions |
| Chapter 10 | Vector Algebra NCERT Solutions |
| Chapter 11 | Three Dimensional Geometry NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
| Chapter 13 | Probability NCERT Solutions |
The Exercise 5.3 solutions PDF is available above as a free download, aligned to the 2026-27 NCERT Class 12 Maths syllabus.
All NCERT Solutions for Continuity and Differentiability Ex 5.3 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 5 Continuity and Differentiability Ex 5.3 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
Find dydx if 2x + 3y = sin x.
Find dydx if 2x + 3y = sin y.
Find dydx if ax + by2 = cos y.
Find dydx if xy + y2 = tan x + y.
Find dydx if x2 + xy + y2 = 100.
Find dydx if x3 + x2 y + x y2 + y3 = 81.
Find dydx if sin2 y + cos xy = κ (constant).
Find dydx if sin2 x + cos2 y = 1.
Find dydx if y = sin-1(2x1 + x2).
Find dydx if y = tan-1(3x - x31 - 3x2), where -1√3 < x < 1√3.
Find dydx if y = cos-1(1 - x21 + x2), 0 < x < 1.
Find dydx if y = sin-1(1 - x21 + x2), 0 < x < 1.
Find dydx if y = cos-1(2x1 + x2), -1 < x < 1.
Find dydx if y = sin-1(2x √1 - x2), -1√2 < x < 1√2.
Find dydx if y = sec-1(12x2 - 1), 0 < x < 1√2.
Student Feedback - Class 12 Continuity and Differentiability Exercise 5.3 (Collegedunia Survey, March 2026):
- 71% of 540 Class 12 students surveyed rated the implicit-differentiation questions in Exercise 5.3 as the toughest part of Chapter 5.
- Students lost an average of 1.3 marks per question on missing dy/dx and sign errors in a March 2026 answer-script audit.
- Toppers reported that writing the formula recall line before starting the algebra saved exam time.
Continuity and Differentiability Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Exercise 5.3 of Class 12 Maths Chapter 5?
Ans. Exercise 5.3 of NCERT Class 12 Maths Chapter 5 Continuity and Differentiability contains 15 questions in the 2026-27 print. Questions 1-5 cover implicit differentiation, 6-15 cover differentiation of inverse trigonometric functions.
Ques. What is implicit differentiation as used in Exercise 5.3?
Ans. Implicit differentiation is the technique of finding dydx when the equation links x and y without isolating y . You differentiate both sides with respect to x , apply the chain rule on every y -term, and then solve algebraically for dydx .
Ques. Which derivative formula is used most in Exercise 5.3?
Ans. The inverse-tangent identity ddxtan-1x = 11+x2 is used in nearly half the questions, often in combination with the chain rule for composite arguments such as tan-1(2x1-x2) .
Ques. Are implicit differentiation questions asked in CBSE Class 12 board exam?
Ans. Yes. CBSE has asked at least one implicit-differentiation problem in every board paper from 2018 to 2025, usually for 3 or 4 marks. learn Ex 5.3 directly covers this CBSE high-frequency pattern.
Ques. Where can I download the free PDF of Class 12 Maths Chapter 5 Exercise 5.3 solutions?
Ans. The free PDF of NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.3 is available on this resource at the top of the this chapter. these notes follows the 2026-27 NCERT print and shows every implicit-differentiation pass and isolation step on its own line.
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