Answer :

Let and.

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

We have

By substituting x = sin θ, we have

[∵ sin^{2}θ + cos^{2}θ = 1]

⇒ = tan^{–1}(tanθ)

Given

However, x = sin θ

Hence, u = tan^{–1}(tanθ) = θ

⇒ u = sin^{–1}x

On differentiating u with respect to x, we get

We know

Now, we have

By substituting x = sin θ, we have

[∵ sin^{2}θ + cos^{2}θ = 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,

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