Updated By SaraNextGen

On March 11, 2024, 11:35 AM

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 = xx

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 xy = u and yx = 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

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