We need to find the derivative of \(\coth(x)\):
\[\frac{d}{dx}\coth(x) = \frac{d}{dx}\frac{\cosh(x)}{\sinh(x)}\]
We can use the qoutient rule:
\[\frac{d}{dx}\coth(x) = \frac{\sinh(x) \frac{d}{dx} \cosh(x) - \cosh(x) \frac{d}{dx} \sinh(x)}{\cosh^2(x)}\]
After the derivation:
\[\frac{d}{dx}\coth(x) = \frac{\sinh^2(x) - \cosh^2(x)}{\sinh^2(x)}\]
Since \(\cosh^2(x) - \sinh^2(x) = 1\):
\[\frac{d}{dx}\coth(x) = \frac{-1}{\sinh^2(x)} = -\operatorname{csch}^2(x) \]