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

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

class 12 maths-contenuety and differentiability

 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.

class 12 maths-contenuety 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 SystemChapter 9-Areas of parallelogram and triangles
Chapter 2-PolynomialChapter 10-Circles
Chapter 3- Coordinate GeometryChapter 11-Construction
Chapter 4- Linear equations in two variablesChapter 12-Heron’s Formula
Chapter 5- Introduction to Euclid’s GeometryChapter 13-Surface Areas and Volumes
Chapter 6-Lines and AnglesChapter 14-Statistics
Chapter 7-TrianglesChapter 15-Probability
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NCERT Solutions for class 9 science 

Chapter 1-Matter in our surroundingsChapter 9- Force and laws of motion
Chapter 2-Is matter around us pure?Chapter 10- Gravitation
Chapter3- Atoms and MoleculesChapter 11- Work and Energy
Chapter 4-Structure of the AtomChapter 12- Sound
Chapter 5-Fundamental unit of lifeChapter 13-Why do we fall ill ?
Chapter 6- TissuesChapter 14- Natural Resources
Chapter 7- Diversity in living organismChapter 15-Improvement in food resources
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NCERT Solutions for class 10 maths

Chapter 1-Real numberChapter 9-Some application of Trigonometry
Chapter 2-PolynomialChapter 10-Circles
Chapter 3-Linear equationsChapter 11- Construction
Chapter 4- Quadratic equationsChapter 12-Area related to circle
Chapter 5-Arithmetic ProgressionChapter 13-Surface areas and Volume
Chapter 6-TriangleChapter 14-Statistics
Chapter 7- Co-ordinate geometryChapter 15-Probability
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Chapter 1- Chemical reactions and equationsChapter 9- Heredity and Evolution
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Chapter 4- Carbon and its CompoundsChapter 12- Electricity
Chapter 5-Periodic classification of elementsChapter 13-Magnetic effect of electric current
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Chapter 7-Control and CoordinationChapter 15-Environment
Chapter 8- How do organisms reproduce?Chapter 16-Management of Natural Resources

NCERT Solutions for class 11 maths

Chapter 1-SetsChapter 9-Sequences and Series
Chapter 2- Relations and functionsChapter 10- Straight Lines
Chapter 3- TrigonometryChapter 11-Conic Sections
Chapter 4-Principle of mathematical inductionChapter 12-Introduction to three Dimensional Geometry
Chapter 5-Complex numbersChapter 13- Limits and Derivatives
Chapter 6- Linear InequalitiesChapter 14-Mathematical Reasoning
Chapter 7- Permutations and CombinationsChapter 15- Statistics
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NCERT solutions for class 12 maths

Chapter 1-Relations and FunctionsChapter 9-Differential Equations
Chapter 2-Inverse Trigonometric FunctionsChapter 10-Vector Algebra
Chapter 3-MatricesChapter 11 – Three Dimensional Geometry
Chapter 4-DeterminantsChapter 12-Linear Programming
Chapter 5- Continuity and DifferentiabilityChapter 13-Probability
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