Differentiate with respect to if

Let and.

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

We have

By substituting 2x = cos θ, we have

[∵ sin²θ + cos²θ = 1]

⇒ u = sin^–1(2 cos θ sin θ)

⇒ u = sin^–1(sin2θ)

Given

However, 2x = cos θ ⇒

Hence, u = sin^–1(sin 2θ) = 2π – 2θ.

⇒ u = 2π – 2cos^–1(2x)

On differentiating u with respect to x, we get

We know and derivative of a constant is 0.

However,

In part (i), we found

We have

Thus,