Question

Which derivative and condition are correct for \[\operatorname{cosec}^{-1}x\]?

Options

• \[\frac{d}{dx}(\operatorname{cosec}^{-1}x)=\frac{1}{|x|\sqrt{x^2-1}},\quad |x|>1\]

• \[\frac{d}{dx}(\operatorname{cosec}^{-1}x)=-\frac{1}{|x|\sqrt{x^2-1}},\quad |x|>1\]

• \[\frac{d}{dx}(\operatorname{cosec}^{-1}x)=-\frac{1}{1+x^2},\quad x\in\mathbb{R}\]

• \[\frac{d}{dx}(\operatorname{cosec}^{-1}x)=-\frac{1}{\sqrt{1-x^2}},\quad |x|<1\]

Answer 1

For \[\operatorname{cosec}^{-1}x\], the denominator involves \[|x|\sqrt{x^2-1}\]. A negative sign is especially important, and \[|x|>1\].