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 = cos θ, we have

[∵ sin²θ + cos²θ = 1]

⇒ u = sin^–1(sinθ)

Given, 0 < x < 1 ⇒ x ϵ (0, 1)

However, x = cos θ

⇒ cos θ ϵ (0, 1)

Hence, u = sin^–1(sinθ) = θ

⇒ u = cos^–1x

On differentiating u with respect to x, we get

We know

Now, we have

By substituting x = cos θ, we have

[∵ sin²θ + cos²θ = 1]

⇒ v = cot^–1(cotθ)

However,

Hence, v = cot^–1(cotθ) = θ

⇒ v = cos^–1x

On differentiating v with respect to x, we get

We know

We have

Thus,