# Derivative of sinh x

How to find the derivative of the given function by using the derivative of sinh x: formula, 1 example, and its solution.

## Hyperbolic Function

The hyperbolic functions are functions

that look like trigonometric functions

with the letter h:

sinh x, cosh x, tanh x, ... .

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

It's read as [hyperbolic sine] or [sin-ch].

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

It's read as [hyperbolic cosine] or [co-sh].

There's a reason

these are called hyperbolic functions.

Recall that

a point (cos θ, sin θ)

is always on the unit circle x^{2} + y^{2} = 1.

Similar to this,

a point ((e^{x} + e^{-x})/2, (e^{x} - e^{-x})/2)

is always on the unit hyperbola x^{2} - y^{2} = 1.

Hyperbola: Equation

So (cosh x, sinh x) is defined as ((e^{x} + e^{-x})/2, (e^{x} - e^{-x})/2).

This is why sinh x = (e^{x} - e^{-x})/2 and cosh x = (e^{x} + e^{-x})/2.

## Formula

The derivative of sinh x, [sinh x]',

is cosh x.

## Exampley = sinh x^{5}, y' = ?

y = sinh x^{5}

is a composite function of

y = sinh (whole) and (whole) = x^{5}.

So the derivative of the composite function is,

the derivative of the outer function sinh x^{5}, cosh x^{5}

times,

the derivative of the inner function x^{5}, 5x^{4}.

Arrange the expression.

So

5x^{4} cosh x^{5}

is the derivative of sinh x^{5}.