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Prove that the derivative of sinh x is cosh x

by RoRi

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

    \[ D (\sinh x) = \cosh x. \]

Proof. We can compute the derivative from the definition of the hyperbolic sine in terms of exponentials,

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

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