The derivative of `sin^(-1)(3x-4x^(3))` with respect to `sin^(-1)x,` is

Options

A. `3," for "|x|lt1`
B. `3," for "|x|lt(1)/(2)" and "-3" for "(1)/(2)lt|x|lt1`
C. `-3," for "|x|lt12`
D. `-3," for "|x|le(1)/(2)" and "3" for "(1)/(2)lt|x|lt1`

Answer

B

Explanation

We have,
`y=sin^(-1)(3x-4x^(3))={{:(3sin^(-1)x" ,"," if "|x|le(1)/(2)),(pi-3sin^(-1)x" ,"," if "(1)/(2)ltxle1),(-pi-3sin^(-1)x","," if "-1lexlt-(1)/(2)):}`
Let`z=sin^(-1)x.` Then,
`y={{:(3z" ,"," if "|x|le(1)/(2)),(pi-3z" ,"," if "(1)/(2)ltxle1),(-pi-3z","," if "-1lexlt-(1)/(2)):}`
`implies" "(dy)/(dz)={{:(3" ,"," if "|x|lt(1)/(2)),(-3","," if "(1)/(2)lt|x|lt1):}`