# Derivative of cosh *x*

How to solve the derivative of cosh *x* problems: formula, proof, example, and its solution.

## Formula

The derivative of cosh *x* is sinh *x*.

Unlike [cos *x*]' = -sin *x*,

the sign doesn't change.

Derivative of cos *x*

## Proof

cosh *x* = (*e*^{x} + *e*^{-x}) / 2

Hyperbolic functions

[*e*^{x}]' = *e*^{x}

Derivative of *e*^{x}

[*e*^{-x}]' = (*e*^{-x})⋅(-1)

Chain rule in differentiation

(*e*^{x} - *e*^{-x}) / 2 = sinh *x*

So [cosh *x*]' = sinh *x*.

## Example

*y* = 1 / (cosh *x*)

So *y*' is equal to,

the derivative of cosh *x*, sinh *x*

over cosh^{2} *x*.

Reciprocal rule in differentiation

sinh *x*/cosh *x* = tanh *x*

1/cosh *x* = sech *x*

So (given) = -tanh *x* sech *x*.