Differentiate with respect to if –1 < x < 1.

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]

⇒ u = sin^–1(2sinθcosθ)

But, sin2θ = 2sinθcosθ

⇒ u = sin^–1(sin2θ)

Given –1 < x < 1 ⇒ x ϵ (–1, 1)

However, x = tan θ

⇒ tan θ ϵ (–1, 1)

Hence, u = sin^–1(sin2θ) = 2θ

⇒ u = 2tan^–1x

On differentiating u with respect to x, we get

We know

Now, we have

By substituting x = tan θ, we have

But,

⇒ v = tan^–1(tan2θ)

However,

Hence, v = tan^–1(tan2θ) = 2θ

⇒ v = 2tan^–1x

On differentiating v with respect to x, we get

We know

We have

Thus,