Q. 95.0( 1 Vote )

# Differentiate <sp

Let and.

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

We have

By substituting x = tan θ, we have

[ sec2θ – tan2θ = 1]

u = sin–1(2sinθcosθ)

But, sin2θ = 2sinθcosθ

u = sin–1(sin2θ)

Given 0 < x < 1 x ϵ (0, 1)

However, x = tan θ

tan θ ϵ (0, 1)

Hence, u = sin–1(sin2θ) = 2θ

u = 2tan–1x

On differentiating u with respect to x, we get

We know

Now, we have

By substituting x = tan θ, we have

[ sec2θ – tan2θ = 1]

v = cos–1(cos2θ – sin2θ)

But, cos2θ = cos2θ – sin2θ

v = cos–1(cos2θ)

However,

Hence, v = cos–1(cos2θ) = 2θ

v = 2tan–1x

On differentiating v with respect to x, we get

We know

We have

Thus,

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