NCERT Solutions class 12 Maths Exercise 5.5

NCERT Solutions class 12 Maths Exercise 5.5 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

Download NCERT solutions for Continuity and Differentiability as PDF.

NCERT Solutions class 12 Continuity & Differentiability

Differentiate the functions with respect to  in Exercise 1 to 5.

1.

Ans. Let   ……….(i)

Taking logs on both sides, we have

 =

 

  

  

  

 

  [From eq. (i)]

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2.

Ans. Let  =   ……….(i)

Taking logs on both sides, we have

  

 

  [From eq. (i)]

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3.  

Ans. Let  ……….(i)

Taking logs on both sides, we have

 

   [By Product rule]

 

 

  =

---

4.

Ans. Let

Putting  and

 

  ……….(i)

Now,

 

 

 

 

 

  =  ……….(ii)

Again, 

      

  

  ……….(iii)

Putting the values from eq. (ii) and (iii) in eq. (i),

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5.  

Ans. Let  …….(i)

Taking logs on both sides, we have

 

 

 

 

 

[From eq. (i)

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Differentiate the functions with respect to  in Exercise 6 to 11.

6.

Ans. Let

Putting             and

  ……….(i)

Now

 

 

 

 

=  ……….(ii)

Again

 

 

 

  ……….(iii)

Putting the values from eq. (ii) and (iii) in eq. (i),

---

7.

Ans. Let  =   where  and

    ……….(i)

Now

 

 

 

 

 

 

  ……….(ii)

Again

 

 

 

 

 

   ……….(iii)

Putting the values from eq. (ii) and (iii) in eq. (i),

 

 

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8.  

Ans. Let  =  where  and

  ……….(i)

Now

 

 

 

 

 

 

   ……….(ii)

Again 

 

  

 

=

=  ……….(iii)

Putting the values from eq. (ii) and (iii) in eq. (i),

---

9.

Ans. Let

Putting  and , we get 

   ……….(i)

Now

  =

 

 

 

 

 

  …..(ii)

Again

  =

 

 

 

 

 

  ……….(iii)

Putting values from eq. (ii) and (iii) in eq. (i),

---

10.

Ans. Let

Putting  and , we have

   ……….(i)

Now   

  =

 

 

 

 

  ……….(ii)

Again 

 

 

 

  ……….(iii)

Putting the values from eq. (ii) and (iii) in eq. (i),

---

11.  

Ans. Let

Putting  and , we have

  ……….(i)

Now

  =

 

 

 

 

  ……….(ii)

Again

  =

 

 

 

 

  ……….(iii)

Putting the values from eq. (ii) and (iii) in eq. (i)

,

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Find  in the following Exercise 12 to 15

12.

Ans. Given:

  where  and

 

  ……….(i)

Now

 

 

 

 

 

 

  ……….(ii)

Again

 

 

 

 

 

 

 ……….(iii)

Putting values from eq. (ii) and (iii) in eq. (i),

 

 

---

13.

Ans. Given:

 

 

 

 

 

 

 

 

---

14.  

Ans. Given:

 

 

 

 

 

 

 

 

 

 

---

15.  

Ans. Given:

 

 

  

 

 

 

 

 

---

16. Find the derivative of the function given by  and hence  

Ans. Given:      ……….(i)

 

 

 

 

 

Putting the value of  from eq. (i),

 

 

  = 8 x 15 = 120

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17. Differentiate  in three ways mentioned below:

(i)    by using product rule.

(ii)   by expanding the product to obtain a single polynomial

(iii)  by logarithmic differentiation.

Do they all give the same answer?

Ans. Let  ……….(i)

(i)

 

 

  ……….(ii)

(ii)

 

 

  ……….(iii)

(iii)

 

 

 

 

 

 

 

 

 

  [From eq. (i)]

  ……….(iv)

From eq. (ii), (iii) and (iv), we can say that value of  is same obtained by three different methods.

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18. If  and  are functions of  then show that  in two ways – first by repeated application of product rule, second by logarithmic differentiation.

Ans. Given:  and  are functions of

To prove:

(i) By repeated application of product rule:

L.H.S.   =

=

=

=

=

=

= R.H.S  Hence proved.

(ii) By Logarithmic differentiation:

Let

 

 

 

 

 

Putting , we get

  Hence proved.

NCERT Solutions class 12 Maths Exercise 5.5

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