## EXERCISE 5.5

#### Page No 178:

#### Question 1:

Differentiate the function with respect to *x*.

#### Answer:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to *x*, we obtain

#### Question 2:

Differentiate the function with respect to *x*.

#### Answer:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to *x*, we obtain

#### Question 3:

Differentiate the function with respect to *x*.

#### Answer:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to *x*, we obtain

#### Question 4:

Differentiate the function with respect to *x*.

#### Answer:

*u *= *x*^{x}

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to *x*, we obtain

*v* = 2^{sin }^{x}

Taking logarithm on both the sides with respect to *x*, we obtain

Differentiating both sides with respect to *x*, we obtain

#### Question 5:

Differentiate the function with respect to *x*.

#### Answer:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to *x*, we obtain

#### Question 6:

Differentiate the function with respect to *x*.

#### Answer:

Differentiating both sides with respect to *x*, we obtain

Differentiating both sides with respect to *x*, we obtain

Therefore, from (1), (2), and (3), we obtain

#### Question 7:

Differentiate the function with respect to *x*.

#### Answer:

*u *= (log *x*)^{x}

Differentiating both sides with respect to *x*, we obtain

Differentiating both sides with respect to *x*, we obtain

Therefore, from (1), (2), and (3), we obtain

#### Question 8:

Differentiate the function with respect to *x*.

#### Answer:

Differentiating both sides with respect to *x*, we obtain

Therefore, from (1), (2), and (3), we obtain

#### Question 9:

Differentiate the function with respect to *x*.

#### Answer:

Differentiating both sides with respect to *x*, we obtain

Differentiating both sides with respect to *x*, we obtain

From (1), (2), and (3), we obtain

#### Question 10:

Differentiate the function with respect to *x*.

#### Answer:

Differentiating both sides with respect to *x*, we obtain

Differentiating both sides with respect to *x*, we obtain

From (1), (2), and (3), we obtain

#### Question 11:

Differentiate the function with respect to *x*.

#### Answer:

Differentiating both sides with respect to *x*, we obtain

Differentiating both sides with respect to *x*, we obtain

From (1), (2), and (3), we obtain

#### Question 12:

Find of function.

#### Answer:

The given function is

Let *x*^{y} = *u* and *y*^{x} = *v*

Then, the function becomes* u *+ *v* = 1

Differentiating both sides with respect to *x*, we obtain

Differentiating both sides with respect to *x*, we obtain

From (1), (2), and (3), we obtain

#### Question 13:

Find of function.

#### Answer:

The given function is

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to *x*, we obtain

#### Question 14:

Find of function.

#### Answer:

The given function is

Taking logarithm on both the sides, we obtain

Differentiating both sides, we obtain

#### Question 15:

Find of function.

#### Answer:

The given function is

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to *x*, we obtain

#### Question 16:

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

#### Answer:

The given relationship is

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to *x*, we obtain

#### Page No 179:

#### Question 18:

If *u*, *v* and *w* are functions of *x*, then show that

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

#### Answer:

Let

By applying product rule, we obtain

By taking logarithm on both sides of the equation, we obtain

Differentiating both sides with respect to *x*, we obtain