Chapter 32
Differentiation of hyperbolic
functions
32.1 Standard differential coefficients
of hyperbolic functions
From Chapter 5,
d
dx
(sinh x) =
d
dx
e x − e −x
2
=
e x − (−e −x )
2
=
e x + e −x
2
= cosh x
If y = sinh ax, where ‘a’ is a constant, then
dy
dx
= a cosh ax
d
dx
(cosh x) =
d
dx
e
x
+ e
−x
2
=
e
x
+ (−e
−x
)
2
=
e x − e −x
2
= sinh x
If y = cosh ax, where ‘a’ is a constant, then
dy
dx
= a sinh ax
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.
Problem 1. Determine the differential coefficient
of: (a) th x (b) sech x.
(a)
d
dx
(th x) =
d
dx
sh x
ch x
=
(ch x)(ch x) − (sh x)(sh x)
ch 2 x
using the quotient rule
=
ch
2 x − sh
2 x
ch 2 x
=
1
ch 2 x
= sech
2 x
(b)
d
dx
(sech x) =
d
dx
1
ch x
=
(ch x)(0) − (1)(sh x)
ch 2 x
=
−sh x
ch 2 x
= −
1
ch x
sh x
ch x
= −sech x th x
Problem 2. Determine
d y
dθ
given
(a) y = cosech θ (b) y = coth θ.
(a)
d
dθ
(cosec θ) =
d
dθ
1
sh θ
=
(sh θ)(0) − (1)(ch θ)
sh
2
θ
=
−ch θ
sh 2 θ
= −
1
sh θ
ch θ
sh θ
= −cosech θ coth θ