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

1.	Prove that `(d)/(dx)(cos^(-1)x)=(1)/(sqrt(1-x^(2))`, where `x in [-1,1].`
Answer» Let `y=cos^(-1)x`, where `x in [-1,1] and y- in [-(pi)/(2),(pi)/(2)].` Then, `y=cos^(-1)x rArr x= cos y` `rArr(dy)/(dx)=-siny," where sin "ygt0," since y"in[0,(pi)/(2)]` `rArr(dy)/(dx)=-sqrt(1-cos^(2)y)=-sqrt(1-x^(2))` `rArr(dy)/(dx)=(-1)/(sqrt(1-x^(2)))` Hence, `(d)/(dx)(cos^(-1)x)=(-1)/(sqrt(1-x^(2)))`.

Discussion

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