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If y=sin^(-1)x+sin^(-1)sqrt(1-x^2) , fin...

If `y=sin^(-1)x+sin^(-1)sqrt(1-x^2)` , find `(dy)/(dx)` in each of the following cases: (i) `x in (0,\ 1)` (ii) `x in (-1,\ 0)`

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`y="sin"^(-1)x+"sin"^(-1)sqrt(1-x^(2))`
Let `x=sin theta`
` :. y=theta+"sin"^(-1)sqrt(1-"sin"^(2))theta=theta+"sin"^(-1)sqrt(cos^(2)theta)`
`=theta+"sin"^(-1)(cos theta)=theta+"sin"^(-1)"sin"((pi)/(2)-theta)`
`=theta +(pi)/(2)-theta=(pi)/(2)`
`implies(dy)/(dx)=(d)/(dx)((pi)/(2))=0`
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NAGEEN PRAKASHAN ENGLISH-Continuity and Differentiability-Miscellaneous Exercise
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  15. If f(x)=|x|^3 , show that f"(x) exists for all real x and find it.

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