Derivative of Hyperbolic Cosine Function

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Theorem

$\map {D_x} {\cosh x} = \sinh x$

where $\cosh$ is the hyperbolic cosine and $\sinh$ is the hyperbolic sine.


Proof

\(\displaystyle \map {D_x} {\cosh x}\) \(=\) \(\displaystyle \map {D_x} {\dfrac {e^x + e ^{-x} } 2}\) Definition of Hyperbolic Cosine
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 2 \map {D_x} {e^x + e^{-x} }\) Derivative of Constant Multiple
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 2 \paren {e^x + \paren {-e^{-x} } }\) Derivative of Exponential Function, Chain Rule for Derivatives, Linear Combination of Derivatives
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {e^x - e^{-x} } 2\)
\(\displaystyle \) \(=\) \(\displaystyle \sinh x\) Definition of Hyperbolic Sine

$\blacksquare$


Also see


Sources