# 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. Differentiate****with respect to x.**

**Solution:**

We have,

On putting x = cot θ, we get,

=

=

= θ

= cot

^{−1 }xDifferentiating with respect to x, we get,

=

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

**Solution:**

We have,,0 < x < ∞

On putting x

^{n}= tan θ, we get,=

Now, 0 < x < ∞

=> 0 < x

^{n}< ∞=> 0 < θ < π/2

=> 0 < 2θ < π

So, y = 2θ

= 2 tan

^{–1}(x^{n})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. Differentiate****with respect to x.**

**Solution:**

We have,

=

Differentiating with respect to x, we get,

= 0 +

=

**Question 26. Differentiate****with respect to x.**

**Solution:**

We have,

=

Differentiating with respect to x, we get,

=

=

**Question 27. Differentiate****with respect to x.**

**Solution:**

We have,

=

=

=

=

Differentiating with respect to x, we get,

= 0 + 1

= 1

**Question 28. Differentiate****with respect to x.**

**Solution:**

We have,

=

=

=

Differentiating with respect to x, we get,

= 0 +

=

**Question 29. Differentiate****with respect to x.**

**Solution:**

We have,

=

=

=

Differentiating with respect to x, we get,

=

=

=

**Question 30. Differentiate****with respect to x.**

**Solution:**

We have,

=

=

Differentiating with respect to x, we get,

=

=

**Question 31. Differentiate****with 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

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