NCERT Solutions Class 12 Maths Chapter 5 guides you through Continuity and Differentiability with clear explanations for all 131 problems across 8 exercises. You’ll learn how to check continuity at a point, apply differentiation rules for composite and implicit functions, use logarithmic differentiation for complex expressions, and find derivatives of inverse trigonometric functions. These techniques are essential for calculus applications in JEE, board exams, and higher mathematics.
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Download PDF (Free)NCERT Solutions Class 12 Maths Chapter 5 Continuity and Differentiability – Complete Guide
NCERT Class 12 Chapter 5 – Continuity and Differentiability forms the backbone of calculus and carries significant weight of 7 marks in your CBSE board examination. This chapter builds upon your Class 11 knowledge of limits and introduces you to two interconnected concepts that are essential for understanding how functions behave. You’ll learn to test functions for continuity at specific points and over intervals, understanding the conditions that make a function continuous and identifying points of discontinuity.
📊 CBSE Class 12 Maths Chapter 5 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 7 Marks |
| Unit Name | Calculus |
| Difficulty Level | Medium |
| Importance | High |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 2-3 questions |
As you progress through this chapter, you’ll discover the precise definition of differentiability and its relationship with continuity. You’ll master powerful differentiation techniques including the chain rule, derivatives of inverse trigonometric functions, implicit differentiation, and parametric differentiation. These tools are crucial not just for board exams but also for competitive examinations like JEE and other engineering entrance tests.
The chapter holds high importance in the CBSE marking scheme, with questions ranging from 2-mark problems on continuity at a point to 4-mark questions on finding derivatives using various methods. You’ll encounter MCQs testing conceptual understanding, short answer questions requiring application of formulas, and long answer problems involving complex composite functions. Understanding logarithmic differentiation and derivatives of functions in parametric form will give you an edge in solving challenging problems.
Quick Facts – Class 12 Chapter 5
| 📖 Chapter Number | Chapter 5 |
| 📚 Chapter Name | Continuity and Differentiability |
| ✏️ Total Exercises | 8 Exercises |
| ❓ Total Questions | 131 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering Continuity and Differentiability opens doors to subsequent chapters like Applications of Derivatives and Integrals. The concepts you learn here have real-world applications in physics (velocity, acceleration), economics (marginal cost, revenue), and engineering. With consistent practice of NCERT solutions and previous year CBSE questions, you’ll develop the analytical skills needed to excel in this mathematically rich and exam-critical chapter.
NCERT Solutions Class 12 Maths Chapter 5 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| Exercise 5.1 | Complete step-by-step solutions for 34 questions | 📥 Download PDF |
| Exercise 5.2 | Complete step-by-step solutions for 10 questions | 📥 Download PDF |
| Exercise 5.3 | Complete step-by-step solutions for 15 questions | 📥 Download PDF |
| Exercise 5.4 | Complete step-by-step solutions for 10 questions | 📥 Download PDF |
| Exercise 5.5 | Complete step-by-step solutions for 18 questions | 📥 Download PDF |
| Exercise 5.6 | Complete step-by-step solutions for 11 questions | 📥 Download PDF |
| Exercise 5.7 | Complete step-by-step solutions for 11 questions | 📥 Download PDF |
| Miscellaneous Exercise on Chapter 5 | Complete step-by-step solutions for 22 questions | 📥 Download PDF |
Continuity and Differentiability – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Continuity at a Point \(\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)\) | Checks if a function f(x) is continuous at x = c. Note: Evaluate left-hand limit, right-hand limit, and the function’s value at the point. All three must be equal for continuity. | When you need to prove or disprove the continuity of a function at a specific point. |
| Chain Rule \(\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)\) | Differentiates a composite function (function inside another function). Note: Differentiate the outer function, keep the inner function as is, then multiply by the derivative of the inner function. | When you have a function within a function, like sin(x²) or e^(cos x). |
| Product Rule \(\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)\) | Differentiates the product of two functions. Note: Derivative of first times second plus first times derivative of second. | When you have two functions multiplied together, like x²sin(x) or e^x log(x). |
| Quotient Rule \(\frac{d}{dx} [\frac{u(x)}{v(x)}] = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}\) | Differentiates the quotient of two functions. Note: Derivative of numerator times denominator minus numerator times derivative of denominator, all divided by denominator squared. Remember the order! | When you have one function divided by another, like sin(x)/x or (x²+1)/(x-1). |
| Derivative of Inverse Function \(\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}\) | Finds the derivative of the inverse of a function. Note: Make sure dy/dx is not equal to zero. | When you need to find dx/dy and you know dy/dx, or when dealing with inverse trigonometric functions. |
| Implicit Differentiation \(\frac{d}{dx} f(y) = f'(y) \cdot \frac{dy}{dx}\) | Differentiates an implicit function (where y is not explicitly defined in terms of x). Note: Differentiate each term with respect to x, remembering to apply the chain rule when differentiating terms involving y. Then, solve for dy/dx. | When you have an equation like x² + y² = 25 where y is not isolated. |
| Derivative of Parametric Function \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\) | Finds dy/dx when x and y are defined in terms of a parameter t. Note: Divide the derivative of y with respect to t by the derivative of x with respect to t. Ensure dx/dt is not zero. | When x = f(t) and y = g(t). |
| Second Order Derivative \(\frac{d^2y}{dx^2} = \frac{d}{dx} (\frac{dy}{dx})\) | Finds the second derivative of a function. Note: First find dy/dx, then differentiate it again with respect to x. | When you need to find the rate of change of the rate of change or to analyze concavity. |
| Logarithmic Differentiation \(\frac{d}{dx} [(\ln y)] = \frac{1}{y} \cdot \frac{dy}{dx}\) | Helps in differentiating functions with variables in both the base and exponent. Note: Take the natural logarithm of both sides of the equation, then differentiate implicitly. Finally, solve for dy/dx. | When you have functions like y = x^x or y = (sin x)^(cos x). |
| L’Hopital’s Rule \(\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\) | Evaluates limits of indeterminate forms like 0/0 or ∞/∞. Note: Differentiate the numerator and denominator separately, then take the limit again. Can be applied multiple times if necessary. Make sure the limit exists. | When direct substitution results in an indeterminate form. |
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