Derivative of Hyperbolic Cosine Function

From ProofWiki
Jump to navigation Jump to search

Theorem

$D_x \left({\cosh x}\right) = \sinh x$

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


Proof

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

$\blacksquare$


Also see


Sources