# NCERT Solutions for Class 12 Maths Chapter 6 – Application of Derivatives

NCERT Solutions for Class 12 Maths Chapter 6 – Application of Derivatives covers all the concepts applying the knowledge of derivatives to real-life situations. In this chapter, you gather knowledge about applications of the derivative in various disciplines like science, social science, and many other fields by learning. You will learn how to determine the rate of change of quantities, how to find the equations of tangent and normal to a curve at a point, how to find turning points on the graph of a function, how to find intervals on which a function is increasing or decreasing and how to find the approximate value of certain quantities.

NCERT Solutions for Class 12 Maths Chapter 6** **has a total of 126 questions divided into 6 exercises which covers all concepts of the chapter in detail. The questions are based on determining the rate of change of quantities, framing equations of tangent along with finding other values from it, and finding approximate values and maxima and minima values within the intervals given.

The solutions for the questions asked in** **NCERT CBSE Class 12 Maths Chapter 6** **are accurate, easy to understand, and drafted in a step by step manner. Expert Maths teachers have drafted the solutions very methodically for you. They help you build a strong conceptual foundation for the board examinations and various entrance examinations that you plan to take after class 12.

## NCERT Class 12 Maths Chapter 6

Our smart, systematic, and strategic NCERT Solutions for Class 12 Maths Chapter 6- Application of Derivatives explain all concepts of the chapter with detailed explanations and relevant solved examples. Some of the concepts covered here have been discussed below.

**Rate of Change of Quantities**

This part of the CBSE NCERT Solutions for Class 12 Maths Chapter 6 teaches you that the rate of change of a quantity *y* with respect to quantity *x* can be calculated using the rate of change of *y* and that of *x* both with respect to *t* and this is given as *dy*/*dx* = (*dy/dt*)/(*dx/dt*) when *dx/dt* ≠ 0. It further tells that, *dy/dx* is positive if *y* increases as *x* increases and is negative if *y* decreases as *x* increases.

**Increasing and Decreasing Functions**

This part of NCERT CBSE Class 12 Maths Chapter 6 teaches you how to use differentiation to find out whether a function is increasing or decreasing, or there is no change.

Further, you will learn how to find out whether a given function is increasing or decreasing or constant, say in a graph. We use derivatives. If *f* is a function which is continuous in [*p*,*q*] and differential on the open interval [*p*,*q*], then; *f* is increasing at [*p*,*q*] if fʹ(*x*) > 0 for each *x* *∈* (*p*,*q*), f is decreasing at [*p*,*q*] if *f*ʹ(*x*) < 0 for each *x* *∈* (*p,q*) and *f* is constant function in [*p*,*q*], if fʹ(*x*) = 0 for each *x* *∈* (*p,q*).

**Tangents and Normals**

This part of the CBSE NCERT Class 12 Maths Book tells you how to use differentiation to find the equation of the tangent line and the normal line to a curve at a given point. Equation of the tangent at (*x*0, *y*0) to the curve *y* = *f *(*x*) is given by *y* – *y*0 = f ʹ(*x _{0}*)(

*x*–

*x*). The equation of the normal to the curve y=

_{0}*f*(

*x*) is given by

*y*–

*y*f ʹ(

_{0}*x*) + (

_{0}*x*–

*x*

_{0}) = 0. This portion mentions some particular cases too.**Approximations**

In this section, you will learn how to use differentials to approximate values of certain quantities. The differential of *x*, denoted by *dx*, is defined by *dx *= Δ*x*. Also, the differential of *y*, denoted by *dy*, is defined by *dy *= *f*ʹ(*x*) *dx *or *dy* = (*dy/dx*) Δ*x.*

**Maxima and Minima**

In this section, you will learn how to use the concept of derivatives to calculate the maximum or minimum values of various functions, the knowledge of such points is very useful in sketching the graph of a given function.

### NCERT CBSE Class 12 Maths Chapter 6 Exercises.

Our NCERT Solutions for Class 12 Maths Chapter 6-Application of Derivatives follow latest CBSE and NCERT curriculum. This chapter comprises 6 exercises, the components of which have been discussed below.

**Exercise 6.1**

This exercise has 18 questions. Questions 1 to 7 asks you to rate of change of parameters of certain geometric figures. Questions 8 to 16 are based on solving the rate of change of parameters of certain daily life objects to find out solutions. Questions 18 and 19 are multiple-choice questions.

**Exercise 6.2**

This exercise has 19 questions. Questions 1, 2, 3, 7, 8, and 9 ask you to show/prove that the given functions are increasing. Questions 4, 5, 6 ask you to show the intervals in which the given functions are increasing/decreasing. The other questions are also based on indicating intervals in which a function is increasing or decreasing.

**Exercise 6.3**

This exercise has 27 questions. Questions 1 to 6 asks you to find slopes of the tangents to given curves. Questions 7 to 9 and 13 ask you to find particular points on given curves. Questions 10 to 12 ask you to find the equation in case the slope of the tangent is given. Questions 14 to 25 are direct questions asking you to find slope, equations of lines, points of a curve. Questions 26 and 27 are multiple type questions.

**Exercise 6.4**

This exercise has 9 questions. Question 1 asks to find the approximate values of the given problems using differentials. Questions 2 and 3 ask you the approximate values of given functions. Questions 4 to 7 are word problems asking to find approximate values of some parameters of geometric figures. Questions 8 to 9 are multiple-choice questions based on the same concept.

**Exercise 6.5**

This exercise has 29 questions. Questions 1 and 2 ask you to find the maximum and minimum values of the given functions. All the subparts of question 3 ask you to determine the local maxima and minima for the expressions given. Subparts of question 4 need to be proved for the absence of maxima and minima. Question 5 asks you to find absolute maxima and minima. Questions 6 to 26 are word problems based on the determination of maximum and minimum values in given situations. Questions 27 to 29 are multiple-choice questions.

**Miscellaneous exercise**

This exercise has 24 questions. These are word problems, and direct questions based on all the concepts learnt in this chapter.

## Benefits of NCERT Solutions for Class 12th Maths Chapter 6 – Application of Derivatives

There are many advantages of studying from our premium quality NCERT Solutions for Class 12 Maths Chapter 6 – Application of Derivatives. Some have been mentioned below.

- They answer all these questions using the best methods of solving these questions.
- They include detailed explanations and relevant examples to help you learn efficiently.
- You can learn anywhere anytime at your own convenience and speed
- Our NCERT Solutions help you keep your concepts clear and help you build a solid conceptual foundation.
- They have been drafted to make learning and revising a faster and more effective process. They have been devised to make learning fun and interesting.