Derivative of Hyperbolic Cosine

From ProofWiki
Jump to navigation Jump to search

Theorem

$\map {\dfrac \d {\d x} } {\cosh x} = \sinh x$

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


Proof

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

$\blacksquare$


Also see


Sources