Question

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

Answer 1

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)))`.