DIFFERENTIATION OF HYPERBOLIC FUNCTIONS 331G
32.2 Further worked problems on
differentiation of hyperbolic
functions
Problem 3. Differentiate the following with
respect tox:(a) y=4sh2x−3
7ch 3x(b) y=5thx
2−2 coth 4x(a) y=4sh2x−
3
7ch 3xdy
dx=4(2 cosh 2x)−3
7(3 sinh 3x)=8 cosh 2x−9
7sinh 3x(b) y=5th
x
2−2 coth 4xdy
dx= 5(
1
2sech^2x
2)
−2(−4 cosech^24 x)=5
2sech^2x
2+8 cosech^24 xProblem 4. Differentiate the following with
respect to the variable: (a) y=4 sin 3tch 4t
(b)y=ln(sh 3θ)−4ch^23 θ.(a) y=4 sin 3tch 4t(i.e. a product)
dy
dx=(4 sin 3t)(4 sh 4t)+(ch 4t)(4)(3 cos 3t)=16 sin 3tsh 4t+12 ch 4tcos 3t
= 4 (4 sin 3tsh 4t+3 cos 3tch 4t)(b) y=ln(sh 3θ)−4ch^23 θ
(i.e. a function of a function)dy
dθ=(
1
sh 3θ)
(3 ch 3θ)−(4)(2 ch 3θ)(3 sh 3θ)=3 coth 3θ−24 ch 3θsh 3θ= 3 (coth 3θ−8ch3θsh 3θ)Problem 5. Show that the differential coeffi-
cient ofy=3 x^2
ch 4xis: 6xsech 4x(1− 2 xth 4x)y=3 x^2
ch 4x(i.e. a quotient)dy
dx=(ch 4x)(6x)−(3x^2 )(4 sh 4x)
(ch 4x)^2=6 x(ch 4x− 2 xsh 4x)
ch^24 x= 6 x[
ch 4x
ch^24 x−2 xsh 4x
ch^24 x]= 6 x[
1
ch 4x− 2 x(
sh 4x
ch 4x)(
1
ch 4x)]= 6 x[sech 4x− 2 xth 4xsech 4x]
= 6 xsech 4x(1− 2 xth 4x)Now try the following exercise.Exercise 135 Further problems on differen-
tiation of hyperbolic functionsIn Problems 1 to 5 differentiate the given func-
tions with respect to the variable:- (a) 3 sh 2[ x (b) 2 ch 5θ (c) 4 th 9t
(a) 6 ch 2x(b) 10 sh 5θ(c) 36 sech^2 9t
]- (a)
2
3sech 5x (b)5
8cosecht
2(c) 2 coth 7θ
⎡⎢
⎢
⎢
⎣(a)−10
3sech 5xth 5x(b)−5
16cosecht
2cotht
2
(c)−14 cosech^27 θ⎤⎥
⎥
⎥
⎦- (a) 2 ln(shx) (b)
3
4ln(
th(
θ
2))[
(a) 2 cothx(b)3
8sechθ
2cosechθ
2]- (a) sh 2xch 2x (b) 3e^2 xth 2x
[
(a) 2(sh^22 x+ch^22 x)
(b) 6e^2 x(sech^22 x+th 2x)]- (a)
3sh4x
2 x^3(b)ch 2t
cos 2t
⎡⎢
⎣(a)12 xch 4x−9sh4x
2 x^4(b)2(cos 2tsh 2t+ch 2tsin 2t)
cos^22 t⎤⎥
⎦