NCERT Solutions Class 12 Maths Chapter 6 guides you through Application of Derivatives with detailed solutions to all 82 questions across 4 exercises. You’ll learn how to find rate of change in real-world problems, determine increasing/decreasing functions, find maxima and minima of functions, and solve optimization problems like maximizing area or minimizing cost. These techniques are crucial for calculus applications in engineering, economics, and physics, plus they carry significant marks in CBSE board exams.
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Download PDF (Free)NCERT Solutions Class 12 Maths Chapter 6 Application of Derivatives – Complete Guide
NCERT Class 12 Chapter 6 – Application of Derivatives transforms the theoretical concept of differentiation into a powerful problem-solving tool. You will explore how derivatives help analyze the behavior of functions, find optimal solutions, and model real-world phenomena. This chapter carries approximately 7 marks in the CBSE board exam and is considered highly important due to its practical applications and frequent appearance in competitive exams like JEE and NEET.
📊 CBSE Class 12 Maths Chapter 6 – 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 |
You’ll begin by understanding rate of change in various contexts – from physics problems involving velocity and acceleration to economics applications like marginal cost and revenue. The chapter then guides you through finding equations of tangents and normals to curves, a concept that appears regularly in 4-mark questions. You’ll also master the approximation technique using differentials, which simplifies complex calculations.
The core of this chapter focuses on determining increasing and decreasing functions using the first derivative test, and finding maxima and minima using both first and second derivative tests. These concepts form the basis for optimization problems – one of the most application-oriented topics in calculus. You’ll solve real-life problems involving maximizing profit, minimizing cost, finding the shortest distance, and optimizing dimensions of geometric shapes.
Quick Facts – Class 12 Chapter 6
| 📖 Chapter Number | Chapter 6 |
| 📚 Chapter Name | Application of Derivatives |
| ✏️ Total Exercises | 4 Exercises |
| ❓ Total Questions | 82 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Expect a mix of question types in your CBSE exam: 2-mark questions on rate of change and tangent-normal equations, 4-mark questions on maxima-minima, and 6-mark application problems requiring complete mathematical reasoning. Building strong problem-solving skills in this chapter will not only boost your board exam score but also strengthen your foundation for higher mathematics and engineering applications. Master the step-by-step approach to optimization problems, and you’ll find this chapter both rewarding and practically valuable.
NCERT Solutions Class 12 Maths Chapter 6 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| Exercise 6.1 | Complete step-by-step solutions for 18 questions | 📥 Download PDF |
| Exercise 6.2 | Complete step-by-step solutions for 19 questions | 📥 Download PDF |
| Exercise 6.3 | Complete step-by-step solutions for 29 questions | 📥 Download PDF |
| Miscellaneous Exercise on Chapter 6 | Complete step-by-step solutions for 16 questions | 📥 Download PDF |
Application of Derivatives – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Rate of Change \(\frac{dy}{dx}\) | Represents the rate of change of y with respect to x. Note: Remember to specify the units of measurement for both variables. | When the problem asks for how fast one quantity is changing compared to another. E.g., rate of change of area with respect to radius. |
| Increasing Function Condition \(\frac{dy}{dx} > 0\) | A function y = f(x) is increasing if its derivative is positive. Note: Strictly increasing. At points where dy/dx = 0, function is neither increasing nor decreasing (stationary point). | To find the intervals where a given function is increasing. |
| Decreasing Function Condition \(\frac{dy}{dx} < 0\) | A function y = f(x) is decreasing if its derivative is negative. Note: Strictly decreasing. At points where dy/dx = 0, function is neither increasing nor decreasing (stationary point). | To find the intervals where a given function is decreasing. |
| Tangent Equation \(y – y_1 = m(x – x_1)\) | Equation of the tangent to the curve y = f(x) at the point (x₁, y₁). Note: m is the slope of the tangent, which is dy/dx evaluated at (x₁, y₁). | To find the equation of a tangent line to a curve at a given point. |
| Normal Equation \(y – y_1 = -\frac{1}{m}(x – x_1)\) | Equation of the normal to the curve y = f(x) at the point (x₁, y₁). Note: The slope of the normal is -1/m, where m is the slope of the tangent (dy/dx evaluated at (x₁, y₁)). | To find the equation of a normal line to a curve at a given point. |
| Slope of Tangent \(m = \frac{dy}{dx}\)_{(x₁, y₁)} | The slope of the tangent line to the curve y = f(x) at the point (x₁, y₁). Note: Evaluate the derivative at the given point (x₁, y₁). | To find the slope of tangent line at a given point on the curve. |
| Approximation using Derivatives \(f(x + \Delta x) \approx f(x) + f'(x) \Delta x\) | Approximates the value of a function at a point slightly different from a known point. Note: \(\Delta x\) is a small change in x. Remember \(f'(x)\) is \(\frac{df}{dx}\). | When you need to find an approximate value of a function at a point close to a known point, especially when direct calculation is difficult. |
| Critical Points (Maxima/Minima) \(\frac{dy}{dx} = 0\) | Points where the derivative of the function is zero. These are potential points of maxima or minima. Note: These points need to be further tested using the first or second derivative test. | To find the points where the function may have a maximum or minimum value. |
| First Derivative Test Sign of \(\frac{dy}{dx}\) changes at critical point | Determines if a critical point is a local maximum or minimum by examining the sign change of the first derivative. Note: If \(\frac{dy}{dx}\) changes from +ve to -ve at x=c, then x=c is a point of local maxima. If \(\frac{dy}{dx}\) changes from -ve to +ve at x=c, then x=c is a point of local minima. | To determine if critical points are local maxima or minima. |
| Second Derivative Test (Maxima) \(\frac{d^2y}{dx^2} < 0\) | If at a critical point, the second derivative is negative, then the function has a local maximum at that point. Note: Easier to use than first derivative test in many cases. f”(x) < 0 implies concave down. | To confirm if a critical point is a point of local maxima. |
| Second Derivative Test (Minima) \(\frac{d^2y}{dx^2} > 0\) | If at a critical point, the second derivative is positive, then the function has a local minimum at that point. Note: Easier to use than first derivative test in many cases. f”(x) > 0 implies concave up. | To confirm if a critical point is a point of local minima. |
| Point of Inflection \(\frac{d^2y}{dx^2} = 0\) and sign change of \(\frac{d^2y}{dx^2}\) | A point where the concavity of the curve changes. Note: The second derivative must change sign at the point of inflection. | To find the point where the graph of a function changes from concave up to concave down or vice versa. |
| Absolute Maxima/Minima Evaluate f(x) at endpoints and critical points | Find the absolute maximum and minimum values of a function on a closed interval [a, b]. Note: Evaluate the function at all critical points within the interval and at the endpoints of the interval. Compare the values. | To determine the largest and smallest value of a function over a specific interval. |
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