Answer
$-\frac{1}{\sqrt{( {({e}^{2t})-1})}}$
Work Step by Step
Given that $y$ = $cosec^{-1}{{e}^{t}}$, on substituting ${{e}^{t}}$ = $u$ and applying the chain rule derivative, we get:
$\frac{dy}{dt}$ = $\frac{{dcosec^{-1}{u}}}{du}$ $\times$ $\frac{du}{dt}$
or $\frac{dy}{dt}$ = $-\frac{1}{{|u|}\sqrt {({u^2}-1)}}$ $\times$ $\frac{du}{dt}$
so $\frac{dy}{dt}$ = $-\frac{1}{|({e}^{t})|\sqrt {({{e}^{2t}}-1})}$ $\times$ $\frac{d {{e}^{t}}}{dt}$
or $\frac{dy}{dt}$ = $-\frac{1}{|{e}^{t}|\sqrt{( {({e}^{2t})-1})}}$ $\times$ ${e}^{t}$
The final answer is: = $-\frac{1}{\sqrt{( {({e}^{2t})-1})}}$
