Prove that `(d)/(dx)(sin^(-1)x)=(1)/(sqrt(1-x^(2))`, where `x in [-1,1

1.	Prove that `(d)/(dx)(sin^(-1)x)=(1)/(sqrt(1-x^(2))`, where `x in [-1,1].`
Answer» Let `y=sin^(-1)x`, where `x in [-1,1] and y- in [-(pi)/(2),(pi)/(2)].` Then, `y=sin^(-1)x rArr x= sin y` `rArr(dx)/(dy)=cosy ge0" since "yin[-(pi)/(2),(pi)/(2)]` `rArr(dx)/(dy)=sqrt(1-sin^(2)y)=sqrt(1-x^(2))` `rArr(dy)/(dx)=(1)/(sqrt(1-x^(2)))` Hence, `(d)/(dx)(sin^(-1)x)=(1)/(sqrt(1-x^(2))).`

Discussion

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