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Prove that the derivative of csch x is -csch x coth x

by RoRi

Prove the following formula for the derivative of the hyperbolic cosecant,

    \[ D(\operatorname{csch} x) = - \operatorname{csch x} \operatorname{coth} x. \]

Proof. We can compute the derivative directly,

    \begin{align*} D(\operatorname{csch} x) &= D \left( \frac{1}{\sinh x} \right) \\ &= \frac{-\cosh x}{\sinh^2 x} \\ &= -\operatorname{csch} x \operatorname{coth} x. \qquad \blacksquare \end{align*}

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