hyperbolic functions problem solving tricks with exam examples questions and answers

3.20 Hyperbolic Functions

The hyperbolic cosine and hyperbolic sine of the real number x are denoted
by coshx and sinhx and are defined to be;
ex +e−x ex −e−x
coshx = 2
and sinhx = 2
sinhx ex −e−x
tanhx = coshx
= ex +e−x
coshx ex +e−x
cothx = sinhx
= ex −e−x
,x 6= 0
1 2
sechx = coshx
= ex +e−x
1 2
cosechx = sinhx
= ex −e−x
,x 6= 0
cosh2 x − sinh2 x = 1
1 − tanh2 x = sech2 x
coth2 x − 1 = cosech2 x
sinhx(x + y) = sinhxcoshy + coshxsinhy
coshx(x + y) = coshxcoshy + sinhxsinhy
sinh2x = 2sinhxcoshx
cosh2x = cosh2 x + sinh2 x




3.21 Derivatives of Hyperbolic Functions
ex +e−x
1. Let y = coshx = 2
dy ex −e−x
dx
= 2

= sinhx

ex −e−x
2. Let y = sinhx = 2
dy ex +e−x
dx
= 2

= coshx

2020

, sinhx
3. Let y = tanhx = coshx
dy (coshx)(coshx)−(sinhx)(sinhx)
∴ dx
= cosh2 x
cosh2 x−sinh2 x
= cosh2 x

= sech2 x

coshx
4. Let y = cothx = sinhx

∴ dy
dx
= (sinhx)(sinhx)−(coshx)(coshx)
sinh2 x
2 2
−(cosh x−sinh x)
= sinh2 x
−1
= sinh2 x

= −cosech2 x

1
5. Let y = sechx = coshx
dy −sinhx
∴ dx
= cosh2 x
−1
= ( coshx )( sinhx
coshx
)
= −sechxtanhx

1
6. Let y = cosechx 
= sinhx
 
dy −coshx −1 coshx
∴ dx
= sinh2 x
= sinhx sinhx

= −cosechxcothx

Examples

1. y = sinh2 x
dy
dx
= 2sinhxcoshx

2. y = cosh(3x − 1)
dy
dx
= 3sinh(3x − 1)

3. y = xsinhx
dy
dx
= (1)sinhx + xcoshx
= sinhx + xcoshx


2020