Answer :

Let and v = sin–1(3x – 4x3)


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


We have


By substituting x = tan θ, we have





Given,


However, x = tan θ





As tan 0 = 0 and tan = 1, we have .


Thus, lies in the range of tan–1x.


Hence,



On differentiating u with respect to x, we get




We know and derivative of a constant is 0.




Now, we have v = sin–1(3x – 4x3)


By substituting x = sin θ, we have


v = sin–1(3sinθ – 4sin3θ)


But, sin3θ = 3sinθ – 4sin3θ


v = sin–1(sin3θ)


Given,


However, x = sin θ





Hence, v = sin–1(sin3θ) = 3θ


v = 3sin–1x


On differentiating v with respect to x, we get




We know




We have





Thus,


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