# Derivative of sinh *x*

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

## Formula

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

## Proof

sinh *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 = cosh *x*

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

## Example

*y* = sinh (*x*^{3})

So *y*' is equal to,

the derivative of the outer part, cosh (*x*^{3})

times, the derivative of the inner part, 3*x*^{2}.

Chain rule in differentiation

Power rule in differentiation (Part 1)