Originally Posted by
SlipEternal $\displaystyle \begin{align*} y' & = \dfrac{ \text{sech}^2\left( \frac{x}{2} \right) }{ 2 \tanh\left( \frac{x}{2} \right) } \\ & = \dfrac{ \left( \dfrac{ 1 }{ \cosh^2\left( \frac{x}{2} \right) } \right) }{ 2 \dfrac{ \sinh\left( \frac{x}{2} \right) }{ \cosh\left( \frac{x}{2} \right) } } \cdot \dfrac{\cosh^2\left( \frac{x}{2} \right) }{ \cosh^2\left( \frac{x}{2} \right) } \\ & = \dfrac{ 1 }{ 2\sinh\left( \frac{x}{2} \right)\cosh\left( \frac{x}{2} \right) } \\ & = \dfrac{1}{\sinh(x)} \\ & = \text{csch}(x) \end{align*}$