NCERT solutions for Class 12 Mathematics Chapter 5 Continuity and Differentiability – complete answers & explanations

Class 12 Mathematics Chapter 5 Continuity and Differentiability is an important chapter that helps students understand how functions behave and change. It introduces key ideas like continuity at a point, differentiability, and how to calculate derivatives of different types of functions. This chapter builds a strong foundation for higher-level calculus and is essential for board exams as well as competitive exams. Students often find this topic slightly challenging, so having clear and reliable NCERT solutions can make a big difference in learning. These solutions are aligned with NCERT answers and help students understand the correct approach to solving each question. Download the worksheet and practice alongside solutions for better clarity. Book a free trial now to get expert guidance and strengthen your understanding of this chapter step by step.

What this NCERT chapter covers?

1. The concept of continuity of functions and how to check whether a function is continuous at a given point.
2. Understanding points of discontinuity and how functions behave at those points.
3. Introduction to differentiability and its relation with continuity.
4. Methods to find derivatives of algebraic, trigonometric, exponential, and logarithmic functions.
5. Use of standard derivative formulas and rules.
6. Implicit differentiation and how to differentiate functions involving more than one variable.
7. Derivatives of inverse trigonometric functions and their applications.
8. Logarithmic differentiation and exponential functions.
9. Higher-order derivatives and their basic understanding.
10. Application-based questions that test conceptual clarity and calculation skills.
11. Importance of step-by-step solving techniques for accuracy in exams.
12. Practice of mixed problems to strengthen overall understanding of continuity and differentiability.

How to use these NCERT solutions?

1. Start by attempting each question from the worksheet on your own without looking at the answers.
2. Carefully compare your answers with the given solutions to check correctness.
3. Identify mistakes and understand where your approach differed from the correct method.
4. Revise the related concept if you find repeated errors in similar questions.
5. Follow the exact order of questions as given to stay consistent with the worksheet structure.
6. Use these solutions as a guide for writing step-by-step answers in exams.
7. Parents and teachers can help students by explaining difficult steps and ensuring regular practice.
8. Focus on understanding the logic behind answers rather than just memorizing them.
9. Practice similar questions to build confidence and improve speed.
10. Revisit tricky questions multiple times to strengthen conceptual clarity.

Important tips & tricks for students

1. Always check continuity by evaluating both left-hand and right-hand limits carefully.
2. Learn standard derivative formulas thoroughly to save time during exams.
3. Avoid skipping steps, especially in differentiation problems.
4. Practice implicit differentiation regularly as it can be tricky.
5. Be careful with signs while working with trigonometric functions.
6. Revise inverse trigonometric derivatives multiple times for accuracy.
7. Focus on understanding the concept instead of rote learning formulas.
8. Attempt all questions, even if unsure, to build confidence.
9. Manage time properly during exams by practicing regularly.
10. Keep revising important formulas and identities to avoid mistakes.

NCERT solutions – complete answer key

Exercise No. 5.1

1. Continuous at x = 0, x = –3, x = 51  
2. Continuous at x = 32  
3. (a) Continuous for all real x  
  (b) Continuous for all x ≠ 5  
  (c) Continuous for all x ≠ –5  
  (d) Continuous for all real x  
4. Continuous at x = n 4  
5. At x = 0 → Continuous  
6. At x = 1 → Continuous  
7. At x = 2 → Continuous  
8. x = 0  
9. x = 0  
10. x = 1  
11. x = 2  
12. x = 1  
13. Continuous everywhere  
14. Not continuous at x = 1  
15. x = –3, x = 3  
16. x = 2 6  
17. a + 1 = 3b  
18. λ = 1  
19. Continuous at x = π  
20. Continuous at x = 1  
21. cos x → Continuous for all real x  
22. cosec x → Continuous where defined (x ≠ nπ)  
23. sec x → Continuous where defined (x ≠ (2n+1)π/2)  
24. cot x → Continuous where defined (x ≠ nπ)  
25. Continuous  
26. k = 3  
27. k = 3/2  
28. k = –1  
29. k = –3  
30. Continuous for all real x  
31. Continuous for all real x  
32. Continuous for all real x  
33. x = –1, x = 0  
34. Continuous for all real x  

Exercise No. 5.2

1. 2x cos(x² + 5)  
2. –cos x sin(sin x)  
3. a cos(ax + b)  
4. c  
5. (a cos(ax + b) cos(cx + d) – c sin(ax + b) sin(cx + d)) / cos²(cx + d)  
6. –2 cot x cosec²x  
7. Not differentiable at x = 1  
8. Not differentiable at x = 1, x = 2  
9. –sin x cos(cos x)  
10. –3x² sin(x³) sin²(x⁵) + 10x⁴ cos(x³) sin(x⁵) cos(x⁵)  

Exercise No. 5.3

1. dy/dx = (cos x – 2)/3  
2. dy/dx = (–a)/(2by – sin y)  
3. dy/dx = (sec²(x + y) – y)/(x + 2y)  
4. dy/dx = (cos y – 2)/3  
5. dy/dx = –(3x² + 2xy + y²)/(x² + 2xy + 3y²)  
6. dy/dx = (–2x – y)/(x + 2y)  
7. dy/dx = (sin x cos x)/(sin y cos y)  
8. dy/dx = (3(1 – x²))/(1 + x²)²  
9. dy/dx = (sin xy·y)/(2 sin y cos y – x sin xy)  
10. dy/dx = (2x)/(1 + x²)  
11. dy/dx = 1/(1 + x²)  
12. dy/dx = (1 – 2x)/(√(1 – (2x – 1)²))  
13. dy/dx = –1/(1 + x²)  
14. dy/dx = –1/(1 + x²)  
15. dy/dx = sec x tan x / √(sec²x – 1)  

Exercise No. 5.4

1. d/dx (ex) = ex  
2. d/dx (ax) = ax log a  
3. d/dx (log x) = 1/x  
4. d/dx (log(sin x)) = cos x / sin x  
5. d/dx (ex sin x) = ex(sin x + cos x)  
6. d/dx (log(ax + b)) = a/(ax + b)  
7. d/dx (x log x) = log x + 1  
8. d/dx (e2x) = 2e2x  
9. d/dx (ex + log x) = ex + 1/x  
10. d/dx (log(ex + 1)) = ex/(ex + 1)  

Exercise No. 5.5

1. d/dx (sin⁻¹ x) = 1/√(1 – x²)  
2. d/dx (cos⁻¹ x) = –1/√(1 – x²)  
3. d/dx (tan⁻¹ x) = 1/(1 + x²)  
4. d/dx (cot⁻¹ x) = –1/(1 + x²)  
5. d/dx (sec⁻¹ x) = 1/(|x|√(x² – 1))  
6. d/dx (tan⁻¹ (x²)) = 2x/(1 + x⁴)  
7. d/dx (sec⁻¹ (2x)) = 2/(|2x|√(4x² – 1))  
8. d/dx (cosec⁻¹ x) = –1/(|x|√(x² – 1))  
9. d/dx (sin⁻¹ (2x)) = 2/√(1 – 4x²)  
10. d/dx (cos⁻¹ (3x)) = –3/√(1 – 9x²)  

Exercise No. 5.6

1. d/dx (sin x + cos x) = cos x – sin x  
2. d/dx (tan x + sec x) = sec²x + sec x tan x  
3. d/dx (cot x + cosec x) = –cosec²x – cosec x cot x  
4. d/dx (sin x cos x) = cos²x – sin²x  
5. d/dx (tan x sec x) = sec³x + sec x tan²x  
6. d/dx (x sin x) = sin x + x cos x  
7. d/dx (x cos x) = cos x – x sin x  
8. d/dx (x tan x) = tan x + x sec²x  
9. d/dx (x log x) = log x + 1  
10. d/dx (x ex) = ex + x ex  
11. dy/dx = –y/x  

Exercise No. 5.7

1. d/dx (sin⁻¹ x + cos⁻¹ x) = 0  
2. d/dx (tan⁻¹ x + cot⁻¹ x) = 0  
3. d/dx (sec⁻¹ x + cosec⁻¹ x) = 0  
4. d/dx (log (sin x)) = cot x  
5. d/dx (log (cos x)) = –tan x  
6. d/dx (log (tan x)) = sec²x / tan x  
7. d/dx (log (sec x)) = tan x  
8. d/dx (log (cosec x)) = –cot x  
9. d/dx (ex log x) = ex log x + ex/x  
10. d/dx (x log x – x) = log x  

MISCELLANEOUS EXERCISE on Chapter 5

1. (3x² – 9x + 5)⁸ (2x – 3)  
2. 3sin²x cos x – 6cos⁵x sin x  
3. 15x² cos 2x – 2(5x)³ sin 2x  
4. 1/√(1 – x²)  
5. (2x(9 – x²) – 2x(1 – cos x)) / (9 – x²)²  
6. (cos x (x + sin x) – sin x (1 + cos x)) / (x + sin x)²  
7. (log x)log x (1/x log x + 1/x)  
8. –sin(a cos x + b sin x)(–a sin x + b cos x)  
9. 2(sin x – cos x)(sin x + cos x)  
10. x^x (log x + 1)  
11. 2x³ + 3x² – 3  
12. 3/5  
13. 0  
14. –(y + 1)/(x + 1)  
15. constant  
16. (2 cos(a + y))/sin a  
17. 1/(1 + x²)  
18. 0  
19. –1/(1 + x²)  
20. 1/√(1 – x²)  
21. 1/(|x|√(x² – 1))  
22. –1/√(1 – x²)  

Why NCERT solutions help students?

NCERT solutions play a key role in helping students prepare effectively for exams by providing clear and structured answers. They improve concept clarity, ensure students follow the correct answering pattern, and build confidence while solving questions. With regular practice, students can strengthen their understanding and perform better in their Class 12 Mathematics exams.

Help your child build strong Mathematics fundamentals with expert-guided learning support.  
Book a free trial!