differentiation

Differential calculus
32
Differentiation of hyperbolic functions
32.1 Standard differential coefficients of
hyperbolic functions
From Chapter 5,
d
dx (sinh x) = d ( e x − e −x ) [ e x − (−e −x ]
)
=
dx 2
2
( e x + e −x )
=
= cosh x
2
If y = sinh ax, where ‘a’ is a constant, then
dy
= a cosh ax
dx
d
dx ( cosh x) = d ( e x + e −x ) [ e x + ( − e −x ]
)
=
dx 2
2
( e x − e −x )
=
= sinh x
2
If y = cosh ax, where ‘a’ is a constant, then
dy
= a sinh ax
dx
Using the quotient rule of differentiation the derivatives
of tanh x, sech x, cosech x and coth x may be
determined using the above results.
(a)
(b)
Problem 1. Determine the differential coefficient
of: (a) th x (b) sech x.
d
dx (th x) = d ( ) sh x
dx ch x
(ch x)(ch x) − (sh x)(sh x)
=
ch 2 x
using the quotient rule
= ch2 x − sh 2 x
ch 2 = 1
x ch 2 x = sech2 x
d
dx (sech x) = d ( ) 1
dx ch x
(ch x)(0) − (1)(sh x)
=
ch 2 x
(a)
(b)
= −sh x ( )( )
1 sh x
ch 2 x =− ch x ch x
= −sech x th x
Problem 2. Determine dy
dθ given
(a) y = cosech θ (b) y = coth θ.
d
dθ (cosec θ) = d ( ) 1
dθ sh θ
d
dθ ( coth θ) = d dθ
(sh θ)(0) − (1)(ch θ)
=
sh 2 θ
= −ch θ
sh 2 θ =− ( 1
sh θ
= −cosech θ coth θ
( ) ch θ
sh θ
)( ) ch θ
sh θ
(sh θ)(sh θ) − (ch θ)(ch θ)
=
sh 2 θ
= sh2 θ − ch 2 θ
sh 2 θ
= −1
sh 2 θ = −cosech2 θ
= −(ch2 θ − sh 2 θ)
sh 2 θ
Summary of differential coefficients
y or f (x)
sinh ax
cosh ax
tanh ax
sech ax
cosech ax
coth ax
dy
dx or f ′ (x)
a cosh ax
a sinh ax
a sech 2 ax
−a sech ax tanh ax
−a cosech ax coth ax
−a cosech 2 ax