Differentiate `tan^(-1)(x/(sqrt(1-x^2)))` with respect to `sin^(-1)(2xsqrt(1-x^2)),` if `-1/(sqrt(2)) < x< 1 /(sqrt(2))`

Explanation

Let `{f}({x})=\tan ^{-1}(\frac{{x}}{\sqrt{1-{x}^{2}}}) and g(x)=\sin ^{-1}(2 {x} \sqrt{1-{x}^{2}})`


Let `x=\sin \theta`



`{f}(\theta)=\tan ^{-1}(\frac{\sin \theta}{\sqrt{1-\sin ^{2} \theta}})=\tan ^{-1}(\frac{\sin \theta}{\cos \theta})=\tan ^{-1}(\tan \theta) `



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