Q. 115.0( 1 Vote )

Differentiate with respect to if

Answer :

Let and.


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


We have


By substituting x = sin θ, we have




[ sin2θ + cos2θ = 1]


u = sin–1(2sinθcosθ)


u = sin–1(sin2θ)


Given


However, x = sin θ





Hence, u = sin–1(sin 2θ) = 2θ.


u = 2sin–1(x)


On differentiating u with respect to x, we get




We know




Now, we have


By substituting x = sin θ, we have




[ sin2θ + cos2θ = 1]



v = tan–1(tanθ)


We have


Hence, v = tan–1(tanθ) = θ


v = sin–1x


On differentiating v with respect to x, we get



We know



We have





Thus,


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