Question

If $y=\text{Cosec}^{-1}(x)$ and $\frac{d y}{d x}=\frac{-1}{|x| \sqrt{x^{2}-1}}$, then

• A. $y \in\left(-\frac{\pi}{2}, 0\right)$
• B. $y \in\left(-\frac{\pi}{2}, 2 \pi\right)$
• C. $y \in\left(-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right)$
• D. $y \in R$

Answer 1

Answer: (C)

It is given that, $y=\text{cosec}^{-1} x$
and $\frac{d y}{d x}=\frac{-1}{|x| \sqrt{x^{2}-1}}$
$\because$ Domain of $\text{cosec}^{-1} x$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\{0\}$
and the $\frac{d y}{d x}$ is define for $x \in\left(-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right)$.
Because for derivatives we should exclude the end points of a internal.