Differentiate `sin^(-1)(2xsqrt(1-x^2))` with respect to `sec^(-1)(1/(sqrt(1-x^2)))` , if `x in (0,\ 1//sqrt(2))`

Explanation

Let `u=\sin ^{-1}(2 x \sqrt{1-x^{2}}) and v=\sec ^{-1}(\frac{1}{\sqrt{1-x^{2}}})`


We need to differentiate `u` with respect to `v` that is find `\frac{d u}{d v}`.



We have `u=\sin ^{-1}(2 x \sqrt{1-x^{2}})`


By substituting `x=\sin \theta`, we have
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