# EXERCISE 5.5 ( Differentiation )

Differentiate the function with respect to (Class 12 ncert solution math exercise 5.5 differentiation)

Question 1:

Solution: Let

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Question 2:Differentiate the function with respect to

Solution: Let

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Question 3: Differentiate the function with respect to

Solution: Let

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Question 4:Differentiate the function with respect to

Solution: Let
Also, let and

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Question 5: Differentiate the function with respect to

Solution: Let

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Question 6: Differentiate the function with respect to

Solution: Let

Also, let and

Then,

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Therefore, from (1), (2) and (3);

Question 7: Differentiate the function with respect to

Solution: Let

Then,

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

&

Therefore, from (1), (2) and (3);

Question 8:Differentiate the function with respect to

Solution: Let

Also, let and

Then,

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Differentiating both sides with respect to , we obtain

Therefore, from (1), (2) and (3);

Question 9:Differentiate the function with respect to

Solution: Let

Also, let and

Then,

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Therefore, from and

Question 10:Differentiate the function with respect to

Solution:Let

Also, let and

Then,

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

&

Therefore, from and (3);

Question 11: Differentiate the function with respect to

Solution: Let
Also, let and

Then,

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Therefore, from (1), (2) and (3);

Question 12: Find of the function

Solution:The given function is

Let, and

Then,

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Now,

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Therefore, from (1), (2) and (3);

Question 13: Find of the function

Solution: The given function is

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Question 14: Find of the function

Solution: The given function is

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Question 15: Find of the function

Solution: The given function is

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Question 16: Find the derivative of the function given by and hence find

Solution: The given function is

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Hence,

Question 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?

Solution: Let

(i) By using product rule

Let and

(ii) By expanding the product to obtain a single polynomial.

Therefore,

(iii) By logarithmic differentiation.

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

From the above three observations, it can be concluded that

all the results of are same.

Question 18:If and are functions of , then show that

in two ways – first by repeated application of product rule, second by logarithmic differentiation.

Solution: Let

By applying product rule, we get

Taking logarithm on both the sides of the equation ,
we obtain,

Differentiating both sides with respect to , we obtain