# Derivative of Hyperbolic Cosine Function

## Theorem

Let $u$ be a differentiable real function of $x$.

Then:

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

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

## Proof

 $\displaystyle \map {\frac \d {\d x} } {\cosh u}$ $=$ $\displaystyle \map {\frac \d {\d u} } {\cosh u} \frac {\d u} {\d x}$ Chain Rule for Derivatives $\displaystyle$ $=$ $\displaystyle \sinh u \frac {\d u} {\d x}$ Derivative of Hyperbolic Cosine

$\blacksquare$