On-line Math 21

3.3.3  Derivatives of inverse hyperbolic functions

If this isn't enough, there are inverses, too. However, from the point of view of calculus, even the trigonometric inverses weren't all that interesting, except that the derivatives were more basic functions. That will be useful when it comes time to find integals of those expressions.

So, inverse hyperbolic functions shouldn't be interesting (in calculus) either, unless they provide new integration formulas. However, they don't (at least, none that can't be easily found other ways). However, there is an odd collection of results: these inverses can be written easily in terms of functions we already know, such as:

sinh-1x = ln(x+ Ö   x2+1   ),

which follows from the definitions, and is explained here.

Now, finding derivatives of sinh^-1x is just an exercise in the chain rule:

( sinh-1x) ¢ = æ è ln æ è x+ Ö   x2+1   ö ø ö ø ¢ = 1 æ è x+ Ö   x2+1   ö ø æ ç ç ç è 1+ x Ö x2+1 ö ÷ ÷ ÷ ø = 1 æ è x+ Ö   x2+1   ö ø æ ç ç ç ç è Ö   x2+1   +x Ö x2+1 ö ÷ ÷ ÷ ÷ ø = 1 Ö x2+1 .

There are similar formulas for the other inverse hyperbolic functions.

Copyright (c) 2000 by David L. Johnson.