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

Question 17. Differentiate, −∞ < x < 0 with respect to x.

Solution:

We have,, −∞ < x < 0

On putting 2^x = tan θ, we get,

=

Now, −∞ < x < 0

=> 0 < 2^x < 1

=> 0 < θ < π/4

=> 0 < 2θ < π/2

So, y = 2θ

= 2 tan⁻¹ (2^x)

Differentiating with respect to x, we get,

=

=

Question 18. Differentiate, a > 1, −∞ < x < 0 with respect to x.

Solution:

We have,, −∞ < x < 0

On putting a^x = tan θ, we get,

=

Now, −∞ < x < 0

=> 0 < a^x < 1

=> 0 < θ < π/4

=> 0 < 2θ < π/2

So, y = 2θ

= 2 tan⁻¹ (a^x)

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 2θ, we get,

=

=

=

=

Now, 0 < x < 1

=> 0 < cos 2θ < 1

=> 0 < 2θ < π/2

=> 0 < θ < π/4

=> π/4 < (θ+π/4) < π/2

So, y =

=

Differentiating with respect to x, we get,

=

=

Question 20. Differentiate, x ≠ 0 with respect to x.

Solution:

We have,

On putting ax = tan θ, we get,

=

=

=

=

=

=

Differentiating with respect to x, we get,

=

Question 21. Differentiate, −π < x < π with respect to x.

Solution:

We have,, −π < x < π

=

=

=

Differentiating with respect to x, we get,

=

Question 22. Differentiatewith respect to x.

Solution:

We have,

On putting x = cot θ, we get,

=

=

= θ

= cot⁻¹ x

Differentiating with respect to x, we get,

=

Question 23. Differentiate, 0 < x < ∞ with respect to x.

Solution:

We have,,0 < x < ∞

On putting xⁿ = tan θ, we get,

=

Now, 0 < x < ∞

=> 0 < xⁿ < ∞

=> 0 < θ < π/2

=> 0 < 2θ < π

So, y = 2θ

= 2 tan^–1 (xⁿ)

Differentiating with respect to x, we get,

=

=

Question 24. Differentiate, x ∈ 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, −π/4 < x < π/4 with respect to x.

Solution:

We have,, −π/4 < x < π/4

=

=

=

=

=

Differentiating with respect to x, we get,

= 0 + 1

= 1