Class 12 Maths NCERT Solutions of exercise 5.2 Chapter- Continuity and Differentiability

 Class 12 maths NCERT  solution of exercise 5.2 -Continuity and differentiability will help all students of 12 class who are studying mathematics. From NCERT  Solutions of exercise 5.2 -Continuity and differentiability class 12 maths textbook you can understand the differentiation of the complex functions like trigonometric functions, rational functions etc.and the derivation of the greatest integer function. On these NCERT solutions of exercise 5.2- Continuity and differentiability, you will get complete knowledge of the chain rule of differentiation. The NCERT Solutions of exercise 5.2 is very easy to understand for the student so students can get the advantage of this by studying exercise 5.2 – Continuity and differentiability.

 

Exercise 5.1-Continuity and Differentiability

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 Class 12 Maths NCERT Solutions of exercise 5.2 Chapter- Continuity and Differentiability

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Exercise 5.2

Differentiate the functions with respect to x in Questions 1 to 8.

Q1.sin( x² + 5)

Let  y = sin( x² + 5)  and t = x² + 5

y = sin t

t = x² + 5

 

= 2x.cos(x² + 5)

Q2. cos(sin x)

Let y = cos(sinx) and t = sin x

y = cos t

t = sin x

Substituting t = sinx

Q3. sin(ax + b)

Let y = sin(ax + b) and t = ax + b

y = sint

t = ax + b

Substituting t = ax + b

Q4.sec(tan√x)

Ans. Let y = sec(tan√x) and t = tan √x

y = sec t

 

t = tan √x

According to the chain rule

Now, we have

 

Substituting the value of t

 

Now evaluating, by chain rule

We have

 

Differentiating the function with respect to x both sides

Let t= x⁵

 

≠

t= x⁵

Similarly, we can evaluate

Therefore

Ans.

Q8. cot (√x)

Let y =cot (√x)

Let √x = t

Substituting value of t

Q9. Prove that the function f  given by

, x ∈ R is not differentiable at x = 1

Ans.

 

It is known that a function f is differentiable at a point x ∈ R in its domain if both

LHS limit = RHS limit

LHS limit

x = 1

LHS limit

 

LHL  ≠ RHL

Therefore f(x) is not differentiable at  x= 1

Q10.Prove that the greatest integer function defined by is not differentiable at x= 1 and  x= 2.

Ans. Given:

 

LHL

 

=

The greatest value of the function at  1-h   im the domain 0 < x < 3  is = 0

= ∞

RHL

 

The greatest value at  1+ h of the function when h tends to 0 is =1

RHL ≠  LHL

Therefore the function is not differentiable at x =1

Let’s check at x = 2

LHL

The value of greatest iteger fuctio [2-h ] = 1

= ∞

RHL

Simce x = 2

The value of greatest fuctio at [ 2 + h] = 2

Therefore f(x) is ot differetiale at x = 2

Himt:  1) If there exist the greatest integer fuctiom [x]   the value of  [1.099] = 1 ,[0.99] = 0 , [2.099] =2, [1.999]=1

[1 +h] = 1 and  [ 1- h] = 0, [2 + h] = 2 and [2-h] where h →0 .

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NCERT Solutions of Science and Maths for Class 9,10,11 and 12

NCERT Solutions for class 9 maths

Chapter 1- Number System	Chapter 9-Areas of parallelogram and triangles
Chapter 2-Polynomial	Chapter 10-Circles
Chapter 3- Coordinate Geometry	Chapter 11-Construction
Chapter 4- Linear equations in two variables	Chapter 12-Heron’s Formula
Chapter 5- Introduction to Euclid’s Geometry	Chapter 13-Surface Areas and Volumes
Chapter 6-Lines and Angles	Chapter 14-Statistics
Chapter 7-Triangles	Chapter 15-Probability
Chapter 8- Quadrilateral

NCERT Solutions for class 9 science 

Chapter 1-Matter in our surroundings	Chapter 9- Force and laws of motion
Chapter 2-Is matter around us pure?	Chapter 10- Gravitation
Chapter3- Atoms and Molecules	Chapter 11- Work and Energy
Chapter 4-Structure of the Atom	Chapter 12- Sound
Chapter 5-Fundamental unit of life	Chapter 13-Why do we fall ill ?
Chapter 6- Tissues	Chapter 14- Natural Resources
Chapter 7- Diversity in living organism	Chapter 15-Improvement in food resources
Chapter 8- Motion	Last years question papers & sample papers

NCERT Solutions for class 10 maths

Chapter 1-Real number	Chapter 9-Some application of Trigonometry
Chapter 2-Polynomial	Chapter 10-Circles
Chapter 3-Linear equations	Chapter 11- Construction
Chapter 4- Quadratic equations	Chapter 12-Area related to circle
Chapter 5-Arithmetic Progression	Chapter 13-Surface areas and Volume
Chapter 6-Triangle	Chapter 14-Statistics
Chapter 7- Co-ordinate geometry	Chapter 15-Probability
Chapter 8-Trigonometry

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Chapter 1- Chemical reactions and equations	Chapter 9- Heredity and Evolution
Chapter 2- Acid, Base and Salt	Chapter 10- Light reflection and refraction
Chapter 3- Metals and Non-Metals	Chapter 11- Human eye and colorful world
Chapter 4- Carbon and its Compounds	Chapter 12- Electricity
Chapter 5-Periodic classification of elements	Chapter 13-Magnetic effect of electric current
Chapter 6- Life Process	Chapter 14-Sources of Energy
Chapter 7-Control and Coordination	Chapter 15-Environment
Chapter 8- How do organisms reproduce?	Chapter 16-Management of Natural Resources

NCERT Solutions for class 11 maths

Chapter 1-Sets	Chapter 9-Sequences and Series
Chapter 2- Relations and functions	Chapter 10- Straight Lines
Chapter 3- Trigonometry	Chapter 11-Conic Sections
Chapter 4-Principle of mathematical induction	Chapter 12-Introduction to three Dimensional Geometry
Chapter 5-Complex numbers	Chapter 13- Limits and Derivatives
Chapter 6- Linear Inequalities	Chapter 14-Mathematical Reasoning
Chapter 7- Permutations and Combinations	Chapter 15- Statistics
Chapter 8- Binomial Theorem	Chapter 16- Probability

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chapter 3-Motion in a Straight Line

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NCERT solutions for class 12 maths

Chapter 1-Relations and Functions	Chapter 9-Differential Equations
Chapter 2-Inverse Trigonometric Functions	Chapter 10-Vector Algebra
Chapter 3-Matrices	Chapter 11 – Three Dimensional Geometry
Chapter 4-Determinants	Chapter 12-Linear Programming
Chapter 5- Continuity and Differentiability	Chapter 13-Probability
Chapter 6- Application of Derivation	CBSE Class 12- Question paper of maths 2021 with solutions
Chapter 7- Integrals
Chapter 8-Application of Integrals

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