Learn Exercise 5.5 with Free Lessons & Tips

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Question 1:

Differentiate the function with respect to x.

Solution 1:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Comments

Swapna Shree

Question 2:

Differentiate the function with respect to x.

Solution 2:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Comments

Swapna Shree

Question 3:

Differentiate the function with respect to x.

Solution 3:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Comments

Swapna Shree

Question 4:

Differentiate the function with respect to x.

Solution 4:

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

Comments

Swapna Shree

Question 5:

Differentiate the function with respect to x.

Solution 5:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Comments

Swapna Shree

Question 6:

Differentiate the function with respect to x.

Solution 6:

Differentiating both sides with respect to x, we obtain

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

Comments

Swapna Shree

Question 7:

Differentiate the function with respect to x.

Solution 7:

u = (log x)^x

Differentiating both sides with respect to x, we obtain

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

Comments

Swapna Shree

Question 8:

Differentiate the function with respect to x.

Solution 8:

Differentiating both sides with respect to x, we obtain

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

Comments

Swapna Shree

Question 9:

Differentiate the function with respect to x.

Solution 9:

Differentiating both sides with respect to x, we obtain

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

Comments

Swapna Shree

Question 10:

Differentiate the function with respect to x.

Solution 10:

Differentiating both sides with respect to x, we obtain

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

Comments

Swapna Shree

Question 11:

Differentiate the function with respect to x.

Solution 11:

Differentiating both sides with respect to x, we obtain

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

Comments

Swapna Shree

Question 12:

Find of function.

Solution 12:

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

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

Comments

Swapna Shree

Question 13:

Find of function.

Solution 13:

Take log both sides and differentiate both sides.

That is xlogy=ylogx

X/y.dy/dx+logy=y.1/x+logx.dy/dx

dy/dx(x/y-logx)=1/x-logy

dy/dx=(1/x-logy)/(x/y-logx)

dy/dx=y(1-xlogy)/x(x-ylogx)

Comments

Vikas Kumar Singh

Question 14:

Find of function.

Solution 14:

The given function is

Taking logarithm on both the sides, we obtain

Differentiating both sides, we obtain

Comments

Swapna Shree

Question 15:

Find of function.

Solution 15:

The given function is

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Comments

Swapna Shree

Question 16:

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

Solution 16:

The given relationship is

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Comments

Swapna Shree

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 17:

(i)

(ii)

(iii)

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

From the above three observations, it can be concluded that all the results of are same.

Comments

Swapna Shree

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.

Solution 18:

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

Comments

Abhay

All Exercises - Chapter 5 - Continuity and Differentiability

Exercise 5.5

Miscellaneous Exercise 5

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Learn Exercise 5.5 with Free Lessons & Tips

All Exercises - Chapter 5 - Continuity and Differentiability

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