Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS

Verify the relation(d/dx)coshx=sinhx.

Using the definition of coshx,

coshx=^12 (ex+e−x),

and differentiating directly, we find

d

dx

(coshx)=^12 (ex−e−x)

=sinhx.

Clearly the integrals of the fundamental hyperbolic functions are also defined

by these relations. The derivatives of the remaining hyperbolic functions can be

derived by product differentiation and are presented below only for complete-

ness.

d

dx

(tanhx)=sech^2 x, (3.54)

d

dx

(sechx)=−sechxtanhx, (3.55)

d

dx

(cosechx)=−cosechxcothx, (3.56)

d

dx

(cothx)=−cosech^2 x. (3.57)

The inverse hyperbolic functions also have derivatives, which are given by the

following:

d

dx

(

cosh−^1

x

a

)

=

1

√

x^2 −a^2

, (3.58)

d

dx

(

sinh−^1

x

a

)

=

1

√

x^2 +a^2

, (3.59)

d

dx

(

tanh−^1

x

a

)

=

a

a^2 −x^2

, forx^2 <a^2 , (3.60)

d

dx

(

coth−^1

x

a

)

=

−a

x^2 −a^2

, forx^2 >a^2. (3.61)

These may be derived from the logarithmic form of the inverse (see subsec-

tion 3.7.5).