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Prove that the derivative of sech x is -sech x tanh x

by RoRi

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

    \[ D(\operatorname{sech} x) = -\operatorname{sech} x \tanh x. \]

Proof. We can compute this directly,

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

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