Differentiate with respect to if 0 < 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

But,

⇒ u = tan^–1(tan2θ)

Given 0 < x < 1 ⇒ x ϵ (0, 1)

However, x = tan θ

⇒ tan θ ϵ (0, 1)

Hence, u = tan^–1(tan2θ) = 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

[∵ sec²θ – tan²θ = 1]

⇒ v = cos^–1(cos²θ – sin²θ)

But, cos2θ = cos²θ – sin²θ

⇒ v = cos^–1(cos2θ)

However,

Hence, v = cos^–1(cos2θ) = 2θ

⇒ v = 2tan^–1x

On differentiating v with respect to x, we get

We know

We have

Thus,