Q. 185.0( 1 Vote )

# Differentiate with respect to if 0 < x < 1.

Answer :

Let and

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

We have

By substituting x = cos θ, we have

[∵ sin^{2}θ + cos^{2}θ = 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^{–1}x

On differentiating u with respect to x, we get

We know

Now, we have

By substituting x = cos θ, we have

[∵ sin^{2}θ + cos^{2}θ = 1]

⇒ v = cot^{–1}(cotθ)

However,

Hence, v = cot^{–1}(cotθ) = θ

⇒ v = cos^{–1}x

On differentiating v with respect to x, we get

We know

We have

Thus,

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