Exercise 5.5

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]

 

 

  = 

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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),

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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),

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9. 

Ans. Let 

Putting  and , we get  

   ……….(i)

Now 

  = 

 

 

 

 

 

  …..(ii)

Again 

  = 

 

 

 

 

 

  ……….(iii)

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

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10. 

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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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),

 

 

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13. 

Ans. Given: 

 

 

 

 

 

 

 

 

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14.  

Ans. Given: 

 

 

 

 

 

 

 

 

 

 

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15.  

Ans. Given: 

 

 

  

 

 

 

 

 

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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.