Differentiate with respect to if –1<x<1, x ≠ 0.  (CBSE 2014,2016)

Let and.

We need to differentiate u with respect to v that is find.

We have

By substituting x = tan θ, we have

[∵ sec²θ – tan²θ = 1]

But, cos2θ = 1 – 2sin²θ and sin2θ = 2sinθcosθ.

Given –1 < x < 1 ⇒ x ϵ (–1, 1)

However, x = tan θ

⇒ tan θ ϵ (–1, 1)

Hence,

On differentiating u with respect to x, we get

We know

Now, we have

By substituting x = tan θ, we have

[∵ sec²θ – tan²θ = 1]

⇒ v = sin^–1(2sinθcosθ)

But, sin2θ = 2sinθcosθ

⇒ v = sin^–1(sin2θ)

However,

Hence, v = sin^–1(sin2θ) = 2θ

⇒ v = 2tan^–1x

On differentiating v with respect to x, we get

We know

We have

Thus,