On-line Math 21

3.3.2  Derivatives of sinh(x) and cosh(x) .

We don't have to use geometric methods to find derivatives of hyperbolic functions; since

sinh(x) : = 1 2 (ex-e-x),

it follows directly that

sinh¢(x) = 1 2 (ex+e-x) = cosh(x),

just as for trig functions. The next rule is a little different, though. Just like with the rules of hyperbolic trigonometry, there are some sign changes between trigonometric and hyperbolic rules:

cosh¢(x) = sinh(x).

Now, the rest is like real trigonometry: tanh(x): = sinh(x)/cosh(x) , sech(x): = 1/cosh(x) , etc..

There are formulas similar to trig functions that do hold:

cosh2x-sinh2x = 1, (sinhx)¢ = coshx, (coshx)¢ = sinhx, (tanhx)¢ = sech2x (sechx)¢ = -sechx tanhx.

In fact, all the rules for derivatives of trigonometric functions have mirror-images as rules for derivatives of hyperbolic functions. The only differences, aside from the ``h'' in the names, is the occasional sign change, like with cosh¢(x) = +sinhx .

Copyright (c) 2000 by David L. Johnson.