NCERT Solutions Class 12

Exercise 5.7

Find the second order derivatives of the functions given in Exercises 1 to 5.

1. 

Ans.  Let 

 

 

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2.  

Ans. Let 

 

 

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3.  

Ans. Let 

  = 

 

= 

= 

= 

= 

= 

---

4.  

Ans. Let 

 

       

  = 

---

5.  

Ans. Let 

 

= 

= 

 

= 

= 

= 

= 

= 

= 

= 

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Find the second order derivatives of the functions given in Exercises 6 to 10.

6.  

Ans. Let 

 

= 

= 

 

= 

= 

= 

= 

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7.  

Ans. Let 

 

= 

=  = 

 

= 

= 

= 

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8.  

Ans. Let 

 

 

= 

=  = 

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9.  

Ans.  Let 

  

= 

 

= 

= 

= 

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10.  

Ans. Let 

 

=  = 

 

= 

= 

= 

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11. If  prove that  

Ans. Let   …….(i)

 

 

=  =  [From eq. (i)]

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12. If  Find  in terms of  alone.

Ans. Given: 

    ……….(i)

 

=  [From eq. (i)]

=  ……….(ii)

 

= 

= 

= 

[From eq. (ii)]

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13. If  show that  

Ans. Given:      ….(i)

 

   = 

  

Now 

 

 

 

   [From eq. (i)]

  Hence proved.

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14. If  show that  

Ans.  Given:     ….(i)

To prove: 

   

          ….(ii)

Again   
=    ….(iii)

Now,   L.H.S.=
= 

= 

= 0

= R.H.S.            Hence proved.

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15. If  show that  

Ans. Given: ……….(i)

  = 

  = 

 

=   [From eq. (i)]

  =   Hence proved.

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NCERT Solutions class 12 Maths Exercise 5.7

16. If  show that  

Ans. Given: 

 

Taking log on both sides,      

 

  = 

And   

 

Now, L.H.S. = 

And R.H.S. = 

 L.H.S. = R.H.S.         Hence proved.

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17. If  show that  

Ans. Given:       ……….(i)

  

And 

= 

  

Again  differentiating both sides w.r.t. 

 

        Hence proved.