Answer
$\displaystyle \frac{d}{dx}[\csc^{-1}u]=-\frac{1}{|u|\sqrt{u^{2}-1}}\cdot\frac{du}{dx},\quad |u|\gt 1$
Work Step by Step
$\displaystyle \csc^{-1}u=\frac{\pi}{2}-\sec^{-1}u\qquad $... differentiate
$\displaystyle \frac{d}{dx}[\csc^{-1}u]=\frac{d}{dx}[\frac{\pi}{2}]-\frac{d}{du}[\sec^{-1}u]\cdot\frac{du}{dx}\quad $
we have derived $\displaystyle \frac{d}{du}[\sec^{-1}u]$ in the text, before example 5.
$\displaystyle \frac{d}{dx}[\csc^{-1}u]=0-\frac{1}{|u|\sqrt{u^{2}-1}}\cdot\frac{du}{dx},\quad |u|\gt 1$
$\displaystyle \frac{d}{dx}[\csc^{-1}u]=-\frac{1}{|u|\sqrt{u^{2}-1}}\cdot\frac{du}{dx},\quad |u|\gt 1$
