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)`


`Rightarrow f(theta)=theta`


`g(theta)=sin ^{-1}(2 sin theta sqrt{1-sin ^{2} theta})=sin ^{-1}(2 sin theta cos theta)=sin ^{-1}(sin 2 theta)`


`Rightarrow g(theta)=2 theta`


`therefore` We have to differentiate `f(theta)` w.r.t. `g(theta)`


`Rightarrow frac{d(f(theta))}{d(g(theta))}=frac{frac{d(f(theta))}{d 3}}{frac{d(g(theta))}{d j}}=frac{frac{d}{d 3}(theta)}{frac{d}{d theta}(2 theta)}=frac{1}{2}`