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

Prove that `(d)/(dx)("cosec"^(-1)x)=(-1)/(|x|sqrt(x^(2)-1))`, where `x

1.	Prove that `(d)/(dx)("cosec"^(-1)x)=(-1)/(\|x\|sqrt(x^(2)-1))`, where `x in R-[-1,1]`.
Answer» Let `y="cosec"^(-1)x`, where `x in R-[-1,1] and y in [-(pi)/(2),(pi)/(2)]-{0}.` Then, `x="cosec y"` `rArr (dx)/(dx)=-"cosec y cot y, where cosec y cot y " gt 0` `rArr (dy)/(dx)=(-1)/("cosec y cot y")=(-1)/(("cosec y")sqrt("cosec"^(2)y-1))=(-1)/(\|x\|sqrt(x^(2)-1)).` Hence, `(d)/(dx)("cosec"^(-1)x)=(-1)/(\|x\|sqrt(x^(2)-1)).`

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