Answer
$\text{coth}^3 v$
Work Step by Step
Since, $\dfrac{d}{dx} (\sinh x)=\cosh x$; $\dfrac{d}{dx} (coth x)=-csch^2 x$ and $coth^2 x -csch^2 x=1$
Now, $\dfrac{dy}{dv} =\dfrac{d}{dv} [\ln \sinh v-\dfrac{1}{2} \text{coth}^2 v)]$
or: $=\dfrac{1}{\sinh v}(\cosh v)-\dfrac{1}{2}(2) \text{coth} v (csch^2 v)$
or: $=\text{coth} v (1+csch^2 v)$
Hence, $\dfrac{dy}{dv}=\text{coth}^3 v$
