Differentiate with respect to if

Let and.

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

We have

By substituting x = sin θ, we have

[∵ sin²θ + cos²θ = 1]

⇒ = tan^–1(tanθ)

Given

However, x = sin θ

Hence, u = tan^–1(tanθ) = θ

⇒ u = sin^–1x

On differentiating u with respect to x, we get

We know

Now, we have

By substituting x = sin θ, we have

[∵ sin²θ + cos²θ = 1]

⇒ v = sin^–1(2sinθcosθ)

⇒ v = sin^–1(sin2θ)

However,

Hence, v = sin^–1(sin 2θ) = 2θ.

⇒ v = 2sin^–1(x)

On differentiating v with respect to x, we get

We know

We have

Thus,