NCERT Class 12 Mathematics Mathematics Part-I: Chapter 6 — APPLICATION OF
This chapter, Application of Derivatives, builds upon the concepts of differentiation learned in Chapter 5. It explores the practical uses of derivatives across various fields like engineering, science, and social science. Key applications include determining the rate of change of quantities, finding equations for tangents and normals to curves, locating local maxima and minima (turning points), identifying intervals where a function is increasing or decreasing, and approximating values of certain quantities. The chapter introduces the concept of rate of change using derivatives, explaining how dy/dx represents the rate of change of y with respect to x. It also demonstrates the use of the Chain Rule for related rates when variables change with respect to a third variable, like time. Examples illustrate calculating the rate of change of a circle's area and the surface area of a cube. This chapter is crucial for understanding how calculus models real-world phenomena and solves complex problems in CBSE Class 12 Mathematics.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 12 |
| Subject | Mathematics |
| Book | Mathematics Part-I |
| Chapter | Chapter 6 — APPLICATION OF |
| Language | English |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 4 minutes |
| Word count | 627 |
Learning outcomes
- Understand derivatives as rates of change for various quantities.
- Apply derivatives to find equations of tangents and normals to curves.
- Determine intervals of increasing and decreasing functions.
- Locate local maxima and minima of functions.
- Use derivatives for approximation of values.
Vocabulary
| Word | Meaning |
|---|---|
| Derivative | The rate at which a function changes with respect to its variable. |
| Rate of change | How one quantity changes in relation to another. |
| Tangent | A line that touches a curve at a single point without crossing it there. |
| Normal | A line perpendicular to the tangent at the point of tangency. |
| Maxima | The maximum value of a function in a given interval or at a point. |
| Minima | The minimum value of a function in a given interval or at a point. |
| Chain Rule | A rule for differentiating composite functions. |
| Approximation | Estimating a value that is close to the true value. |
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Practice questions
- What does dy/dx represent when y is a function of x? Answer: dy/dx represents the rate of change of y with respect to x.
- How is the Chain Rule used when two variables change with respect to a third variable? Answer: The Chain Rule dy/dx = (dy/dt) / (dx/dt) is used to find the rate of change of y with respect to x when both y and x are functions of t.
- If the volume of a sphere is increasing at a rate of 5 cm³/s, find the rate of change of its radius when the radius is 10 cm. Answer: V = (4/3)πr³. dV/dt = 4πr² (dr/dt). Given dV/dt = 5, when r = 10, 5 = 4π(10)² (dr/dt) => dr/dt = 5 / (400π) = 1 / (80π) cm/s.
Practice MCQs
Q1. What does the derivative f'(x) represent at a point x=x₀?
Explanation: The derivative f'(x) at a point x₀ represents the instantaneous rate of change of the function y = f(x) with respect to x at that specific point x₀.
Q2. If x and y are functions of t, and dx/dt ≠ 0, what is the formula for dy/dx using the Chain Rule?
Explanation: The Chain Rule states that if y is a function of t and x is a function of t, then the rate of change of y with respect to x is the ratio of the rate of change of y with respect to t to the rate of change of x with respect to t, provided dx/dt is not zero.
Q3. In Example 1, what is the rate of change of the area of a circle with respect to its radius when r = 5 cm?
Explanation: The area A = πr². The rate of change of area with respect to radius is dA/dr = 2πr. When r = 5 cm, dA/dr = 2π(5) = 10π cm²/s.
Q4. In Example 2, what is given as the rate of increase of the volume of a cube?
Explanation: The problem statement explicitly mentions that the volume of the cube is increasing at a rate of 9 cubic centimetres per second (9 cm³/s).
Q5. The derivative of a function can be used to find:
Explanation: One of the primary applications of derivatives is to determine the rate at which one quantity changes with respect to another.
Frequently asked questions
What is the main purpose of studying the 'Application of Derivatives' in Class 12 Mathematics?
This chapter teaches how derivatives can be used to solve real-world problems, such as finding rates of change, optimizing values, and analyzing function behavior.
How does a derivative represent the rate of change?
The derivative dy/dx of a function y = f(x) represents the instantaneous rate at which y changes with respect to x.
What is the significance of the Chain Rule in this chapter?
The Chain Rule is crucial for finding the rate of change of one variable with respect to another when both are dependent on a third variable (related rates).
Can derivatives be used to find maximum or minimum values of a function?
Yes, derivatives help in finding turning points on a function's graph, which correspond to local maximum or minimum values.
What is the relationship between derivatives and the tangent to a curve?
The derivative of a function at a point gives the slope of the tangent line to the curve at that point.
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NCERT Class 12 Mathematics — Mathematics Part-I — Chapter 6 — APPLICATION OF. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.