Differentiate the following functions with respect to `"sin"(2sin^(-1)x)`

Explanation

We need to find derivative of `sin (2 sin ^{-1} x)`


Apply the chain rule,


Let `f=sin (u)`


Therefore, `frac{d}{d x}(sin (2 arcsin (x)))=frac{d}{d u}(sin (u)) frac{d}{d x}(2 arcsin (x))`


We know that `frac{d}{d u}(sin (u))=cos (u)` and `frac{d}{d x}(2 arcsin (x))=2 frac{d}{d x}(arcsin (x))=2`.


`frac{1}{sqrt{1-x^{2}}}`


So, `frac{d}{d x}(sin (2 arcsin (x)))`


`=frac{d}{d u}(sin (u)) frac{d}{d x}(2 arcsin (x))`


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