# Class 12 RD Sharma Solutions – Chapter 11 Differentiation – Exercise 11.3 | Set 2

Last Updated : 08 May, 2021

### Question 17. Differentiate, âˆ’âˆž < x < 0 with respect to x.

Solution:

We have,, âˆ’âˆž < x < 0

On putting 2x = tan Î¸, we get,

=

Now, âˆ’âˆž < x < 0

=> 0 < 2x < 1

=> 0 < Î¸ < Ï€/4

=> 0 < 2Î¸ < Ï€/2

So, y = 2Î¸

= 2 tanâˆ’1 (2x)

Differentiating with respect to x, we get,

=

=

### Question 18. Differentiate, a > 1, âˆ’âˆž < x < 0 with respect to x.

Solution:

We have,, âˆ’âˆž < x < 0

On putting ax = tan Î¸, we get,

=

Now, âˆ’âˆž < x < 0

=> 0 < ax < 1

=> 0 < Î¸ < Ï€/4

=> 0 < 2Î¸ < Ï€/2

So, y = 2Î¸

= 2 tanâˆ’1 (ax)

Differentiating with respect to x, we get,

=

=

### Question 19. Differentiate, 0 < x < 1 with respect to x.

Solution:

We have,, 0 < x < 1

On putting x = cos 2Î¸, we get,

=

=

=

=

Now, 0 < x < 1

=> 0 < cos 2Î¸ < 1

=> 0 < 2Î¸ < Ï€/2

=> 0 < Î¸ < Ï€/4

=> Ï€/4 < (Î¸+Ï€/4) < Ï€/2

So, y =

=

Differentiating with respect to x, we get,

=

=

### Question 20. Differentiate, x â‰  0 with respect to x.

Solution:

We have,

On putting ax = tan Î¸, we get,

=

=

=

=

=

=

Differentiating with respect to x, we get,

=

### Question 21. Differentiate, âˆ’Ï€ < x < Ï€ with respect to x.

Solution:

We have,, âˆ’Ï€ < x < Ï€

=

=

=

Differentiating with respect to x, we get,

=

### Question 22. Differentiatewith respect to x.

Solution:

We have,

On putting x = cot Î¸, we get,

=

=

= Î¸

= cotâˆ’1 x

Differentiating with respect to x, we get,

=

### Question 23. Differentiate, 0 < x < âˆž with respect to x.

Solution:

We have,,0 < x < âˆž

On putting xn = tan Î¸, we get,

=

Now, 0 < x < âˆž

=> 0 < xn < âˆž

=> 0 < Î¸ < Ï€/2

=> 0 < 2Î¸ < Ï€

So, y = 2Î¸

= 2 tanâ€“1 (xn)

Differentiating with respect to x, we get,

=

=

### Question 24. Differentiate, x âˆˆ R with respect to x.

Solution:

We have,

=

=

Differentiating with respect to x, we get,

= 0

### Question 25. Differentiatewith respect to x.

Solution:

We have,

=

Differentiating with respect to x, we get,

= 0 +

=

### Question 26. Differentiatewith respect to x.

Solution:

We have,

=

Differentiating with respect to x, we get,

=

=

### Question 27. Differentiatewith respect to x.

Solution:

We have,

=

=

=

=

Differentiating with respect to x, we get,

= 0 + 1

= 1

### Question 28. Differentiatewith respect to x.

Solution:

We have,

=

=

=

Differentiating with respect to x, we get,

= 0 +

=

### Question 29. Differentiatewith respect to x.

Solution:

We have,

=

=

=

Differentiating with respect to x, we get,

=

=

=

### Question 30. Differentiatewith respect to x.

Solution:

We have,

=

=

Differentiating with respect to x, we get,

=

=

### Question 31. Differentiatewith respect to x.

Solution:

We have,

=

=

Differentiating with respect to x, we get,

=

=

### Question 32. Differentiate, âˆ’Ï€/4 < x < Ï€/4 with respect to x.

Solution:

We have,, âˆ’Ï€/4 < x < Ï€/4

=

=

=

=

=

Differentiating with respect to x, we get,

= 0 + 1

= 1

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