Answer
$-\frac{1}{|\frac{x}{2}|\sqrt( {( {x^2})-4})}$
Work Step by Step
Given that $y$ = $cosec^{-1}{\frac{x}{2}}$
on substituting ${\frac{x}{2}}$ = $u$
on applying chain rule derivative
$\frac{dy}{dx}$ = $\frac{{dcosec^{-1}{u}}}{du}$ $\times$ $\frac{du}{dx}$
or $\frac{dy}{dx}$ = $-\frac{1}{{|u|}\sqrt ({u^2}-1)}$ $\times$ $\frac{du}{dx}$
so $\frac{dy}{dx}$ = $-\frac{1}{|(\frac{x}{2})|\sqrt ({({\frac{x}{2}})^2-1})}$ $\times$ $\frac{d {(\frac{x}{2})}}{dx}$
or $\frac{dy}{dx}$ = $-\frac{1}{|\frac{x}{2}|\sqrt( {(\frac{x^2}{4})-1})}$ $\times$ $\frac{1}{2}$
The final answer is: $-\frac{1}{|\frac{x}{2}|\sqrt( {( {x^2})-4})}$
