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




[ sin2θ + cos2θ = 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–1x


On differentiating u with respect to x, we get



We know



Now, we have


By substituting x = cos θ, we have




[ sin2θ + cos2θ = 1]



v = cot–1(cotθ)


However,


Hence, v = cot–1(cotθ) = θ


v = cos–1x


On differentiating v with respect to x, we get



We know



We have





Thus,


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