We define:

Note that:

– so an odd function.

– so an even function.

Also: as .

Note:

##### Derivative of sinh and cosh

Exercise

Find derivatives of tanh, coth, sech,cosech.

##### Inverse functions

sinh is an injection and so we have an inverse function denoted by .

Domain = range of

Range = domain of

means the real number whose is x.

tanh is an injection and so we have an inverse function denoted by .

Domain = range of

Range = domain of

means the real number whose is x.

Note that is not an injection, so we need a cut down domain to non-negative x’s () to make it one.

means the non-negative real number whose is x.

##### Derivatives of the inverse hyperbolic functions

.