# Derivative of Hyperbolic Cosine Function

## 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

$\blacksquare$