# NCERT Solutions for Miscellaneous Exercise Chapter 5 Class 12 - Continuity and Differentiability

In the Class 12 Maths chapter 5 miscellaneous exercise solutions, you will get a mixture of questions from all the previous exercises of this Class 12 Maths NCERT textbook chapter. You will get questions related to first-order derivatives of different types of functions, second-order derivatives, mean-value theorem, Rolle's theorem in the miscellaneous exercise chapter 5 Class 12.

This Class 12 NCERT syllabus exercise is a bit difficult as compared to previous exercises, so you may not be able to solve NCERT problems from this exercise at first. You can take help from NCERT solutions for Class 12 Maths chapter 5 miscellaneous exercise to get clarity. There are not many questions asked in the board exams from this exercise, but Class 12 Maths chapter 5 miscellaneous solutions are important for the students who are preparing for competitive exams like JEE main, SRMJEE, VITEEE, MET, etc. You can click on the given link if you are looking for NCERT Solutions for Class 6 to Class 12 at one place.

**Also, see**

- Continuity and Differentiability Exercise 5.1
- Continuity and Differentiability Exercise 5.2
- Continuity and Differentiability Exercise 5.3
- Continuity and Differentiability Exercise 5.4
- Continuity and Differentiability Exercise 5.5
- Continuity and Differentiability Exercise 5.6
- Continuity and Differentiability Exercise 5.7
- Continuity and Differentiability Exercise 5.8

**Continuity and Differentiability Miscellaneous Exercise:**

**Question:1 **Differentiate w.r.t. x the function in Exercises 1 to 11.

**Answer:**

Given function is

Now, differentiation w.r.t. x is

Therefore, differentiation w.r.t. x is

**Question:2 **Differentiate w.r.t. x the function in Exercises 1 to 11.

**Answer:**

Given function is

Now, differentiation w.r.t. x is

Therefore, differentiation w.r.t. x is

**Question:3 **Differentiate w.r.t. x the function in Exercises 1 to 11.

**Answer:**

Given function is

Take, log on both the sides

Now, differentiation w.r.t. x is

By using product rule

Therefore, differentiation w.r.t. x is

**Question:4 ** Differentiate w.r.t. x the function in Exercises 1 to 11.

**Answer:**

Given function is

Now, differentiation w.r.t. x is

Therefore, differentiation w.r.t. x is

**Question:5 **Differentiate w.r.t. x the function in Exercises 1 to 11.

**Answer:**

Given function is

Now, differentiation w.r.t. x is

By using the Quotient rule

Therefore, differentiation w.r.t. x is

**Question:6 **Differentiate w.r.t. x the function in Exercises 1 to 11.

**Answer:**

Given function is

Now, rationalize the [] part

Given function reduces to

Now, differentiation w.r.t. x is

Therefore, differentiation w.r.t. x is

**Question:7** Differentiate w.r.t. x the function in Exercises 1 to 11.

**Answer:**

Given function is

Take log on both sides

Now, differentiate w.r.t.

Therefore, differentiation w.r.t x is

**Question:8** , for some constant a and b.

**Answer:**

Given function is

Now, differentiation w.r.t x

Therefore, differentiation w.r.t x

**Question:9**** **

**Answer:**

Given function is

Take log on both the sides

Now, differentiate w.r.t. x

Therefore, differentiation w.r.t x is

**Question:10** , for some fixed a > 0 and x > 0

**Answer:**

Given function is

Lets take

Now, take log on both sides

Now, differentiate w.r.t x

-(i)

Similarly, take

take log on both the sides

Now, differentiate w.r.t x

-(ii)

Similarly, take

take log on both the sides

Now, differentiate w.r.t x

-(iii)

Similarly, take

take log on both the sides

Now, differentiate w.r.t x

-(iv)

Now,

Put values from equation (i) , (ii) ,(iii) and (iv)

Therefore, differentiation w.r.t. x is

**Question: 11**

**Answer:**

Given function is

take

Now, take log on both the sides

Now, differentiate w.r.t x

-(i)

Similarly,

take

Now, take log on both the sides

Now, differentiate w.r.t x

-(ii)

Now

Put the value from equation (i) and (ii)

Therefore, differentiation w.r.t x is

**Question:12** Find dy/dx if

**Answer:**

Given equations are

Now, differentiate both y and x w.r.t t independently

And

Now

Therefore, differentiation w.r.t x is

**Question:13** Find dy/dx if

**Answer:**

Given function is

Now, differentiatiate w.r.t. x

Therefore, differentiatiate w.r.t. x is 0

**Question:14** If

**Answer:**

Given function is

Now, squaring both sides

Now, differentiate w.r.t. x is

Hence proved

**Question:15** If , for some c > 0, prove that is a constant independent of a and b.

**Answer:**

Given function is

- (i)

Now, differentiate w.r.t. x

-(ii)

Now, the second derivative

Now, put values from equation (i) and (ii)

Now,

Which is independent of a and b

Hence proved

**Question:16** If , with , prove that

**Answer:**

Given function is

Now, Differentiate w.r.t x

Hence proved

**Question:17 ** If and find

**Answer:**

Given functions are

and

Now, differentiate both the functions w.r.t. t independently

We get

Similarly,

Now,

Now, the second derivative

Therefore,

**Question:18** If, show that f ''(x) exists for all real x and find it.

**Answer:**

Given function is

Now, differentiate in both the cases

And

In both, the cases f ''(x) exist

Hence, we can say that f ''(x) exists for all real x

and values are

**Question:19** Using mathematical induction prove that for all positive integers n.

**Answer:**

Given equation is

We need to show that for all positive integers n

Now,

For ( n = 1)

Hence, true for n = 1

For (n = k)

Hence, true for n = k

For ( n = k+1)

Hence, (n = k+1) is true whenever (n = k) is true

Therefore, by the principle of mathematical induction we can say that is true for all positive integers n

**Question:20 **Using the fact that and the differentiation,

obtain the sum formula for cosines.

**Answer:**

Given function is

Now, differentiate w.r.t. x

Hence, we get the formula by differentiation of sin(A + B)

**Question:21 ** Does there exist a function which is continuous everywhere but not differentiable

at exactly two points? Justify your answer.

**Answer:**

Consider f(x) = |x| + |x +1|

We know that modulus functions are continuous everywhere and sum of two continuous function is also a continuous function

Therefore, our function f(x) is continuous

Now,

If Lets differentiability of our function at x = 0 and x= -1

L.H.D. at x = 0

R.H.L. at x = 0

R.H.L. is not equal to L.H.L.

Hence.at x = 0 is the function is not differentiable

Now, Similarly

R.H.L. at x = -1

L.H.L. at x = -1

L.H.L. is not equal to R.H.L, so not differentiable at x=-1

Hence, exactly two points where it is not differentiable

**Question:22 ** If , prove that

**Answer:**

Given that

We can rewrite it as

Now, differentiate w.r.t x

we will get

Hence proved

**Question:23** If , show that

**Answer:**

Given function is

Now, differentiate w.r.t x

we will get

-(i)

Now, again differentiate w.r.t x

-(ii)

Now, we need to show that

Put the values from equation (i) and (ii)

Hence proved

**More About NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous Exercise**

In Class 12 Maths chapter 5 miscellaneous solutions there are 23 questions related to finding derivatives of different types of functions, second-order derivatives, and mixed concepts questions from all the previous exercises of this chapter. Before solving the exercises questions, you can try to solve solved examples given before this exercise. It will help you to get more clarity of the concept and you will be able to solve miscellaneous questions by yourself.

**Also Read|** Continuity and Differentiability Class 12 Chapter 5 Notes

**Benefits of ****NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous Exercise**

**Benefits of**

**NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous Exercise**

- Sometimes questions from miscellaneous exercises are asked in the competitive exam so, the Class 12 Maths chapter 5 miscellaneous solutions becomes important.
- First, try to solve NCERT problems by yourself, it will check your understanding of the concept
- NCERT solutions for Class 12 Maths chapter 5 miscellaneous exercise can be used for reference.

**Also see-**

NCERT Solutions for Class 12 Maths Chapter 5

NCERT Exemplar Solutions Class 12 Maths Chapter 5

**NCERT Solutions of Class 12 Subject Wise**

NCERT Solutions for Class 12 Maths

NCERT Solutions for Class 12 Physics

NCERT Solutions for Class 12 Chemistry

NCERT Solutions for Class 12 Biology

**Subject Wise NCERT Exampler Solutions**

NCERT Exemplar Solutions for Class 12 Maths

NCERT Exemplar Solutions for Class 12 Physics

NCERT Exemplar Solutions for Class 12 Chemistry

NCERT Exemplar Solutions for Class 12 Biology

**Happy learning!!!**

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