Q. 95.0( 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 = 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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