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Prove that the derivative of coth x is -csch2 x

by RoRi

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

    \[ D(\operatorname{coth} x) = -\operatorname{csch}^2 x. \]

Proof. We know from the previous two exercises (here and here that

    \[ D(\sinh x) = \cosh x \qquad \text{and} \qquad D(\cosh x) = \sinh x. \]

Furthermore, we know from this exercise that

    \[ \cosh^2 x - \sinh^2 x = 1 \quad \implies \quad \sinh^2 x - \cosh^2 x = -1. \]

So, we can compute the derivative

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

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