Inverse Hyperbolic Functions
The Inverse Hyperbolic Sine, Inverse Hyperbolic Cosine & Inverse
Hyperbolic Tangent
Rfrangefdom
xxgfunctiontheofiverseThe
hxxf
RfrangeRfdom
xxgfunctiontheofiverseThe
hxxf
frangefdom
xxxgfunctiontheofiverseThe
hxxf
,)1,1(
tanh)(:
arctan)(.3
,
sinh)(:
arcsin)(.2
),0[,),1[
0;cosh)(:
arccos)(.1
The Inverse Hyperbolic Cotangent, Inverse Hyperbolic Secant & Inverse
Hyperbolic Cosecant
}0{,}0{
0csc)(:
csc)(.3
}0{,),1()1,(
0,coth)(:
coth)(.2
),0[,]1,0(
0;sec)(:
sec)(.1
RfrangeRfdom
xhxxgfunctiontheofiverseThe
hxarcxf
RfrangeUfdom
xxxgfunctiontheofiverseThe
xarcxf
frangefdom
xhxxgfunctiontheofiverseThe
hxarcxf
Derivatives of Inverse Hyperbolic Functions
1
1)(arcsin.2
),1(;1
1)(arccos.1
2
2
xh
xx
hx
)1,1(;1
1)(arctan.3
2
x
xhx
)1,0(;1
1)sec(.4
2
x
xxhxarc
),1()1,(;1
1)coth(.5
2
Ux
xxarc
0;1
1
)csc(.6
2
;1
1
;1
1{2
2
xxx
hxarcpositiveisx
xx
positiveisxxx
Proofs
),1(;1
1
sin
1
1sin
1sinh,1;cosh
arccos
:.1
2
2
xxy
y
yy
xysoandxxy
hxy
Let
1
1
cosh
1
1cosh
cosh
1coshsinh
arcsin
:.2
2
2
xyy
yy
negativeneverisybecauserootpositivethechooseWe
xysoandxy
hxy
Let
)1,1(;1
1
sec
1
1sec,)1,1(;tanh
arctan
:.3
22
22
xxhy
y
xhysoandxxy
hxy
Let
)1,0(;1
1
tanhsec
1
1tanhsec
.inttanh),1,0(
sec,
1tanh
1tanh,],1,0(,sec
sec.4
2
2
22
xxxyhy
y
yyhy
ervalthatonpositiveisysoandx
positiveishxarcybecauserootpositivethechooseWe
xy
xysoandxxhy
hxarcy
),1()1,(;1
1
1
1
csc
1
1csc
1csc,),1()1,(;coth
coth
:.5
222
2
22
Uxxxhy
y
yh
xhysoandUxxy
xarcy
Let
0;1
1
tanhsec
1
1cotcsc
coth
1coth,,0,csc
csc.6
2
;1
1
;1
1
;1
;1
22
{
{
2
2
2
2
xxx
yhyy
yyhy
y
xysoandxxy
hxarcy
positiveisxxx
negativeisxxx
positiveisxx
negativeisxx
x
hx
hxy
xhy
hxy
ey
xhy
xhy
casesfollowingtheofeachforyFind
)(arcsin.6
)ln(lnarcsin.5
)ln(arctan.4
.3
)(cosarcsin.2
)3tan5(arccos.1
5
arcsin
25
4
Example (1)
13tan25
sec15
13tan25
3sec5
)3tan5(arccos
2
2
2
2
x
x
x
xy
xhxy
Example (2)
1cos
)(cosarcsin2sin5
1cos
)sin(cos2)(cosarcsin5
)(cosarcsin
4
24
4
24
25
x
xhx
x
xxxhy
xhy
Example (3)
1,1;
1
4
)ln(arctan
1
)ln(arctan
5
10
4
5
5
xlyequivalentorx
x
x
xy
hxy
Integrals Involving Inverse Hyperbolic Functions
}0{;csc1
.6
)1,0(;sec1
.5
),1()1,(;coth1
.4
)1,1(;arctan1
.3
),1(;arccos1
.2
arcsin1
.1
2
2
2
2
2
2
Rxcxharcxx
dx
xcxharcxx
dx
Uxcxarcx
dx
xchxx
dx
xchxx
dx
chxx
dx
Example (1)
ch
dxx
dxx
x
dxx
x
dxx
x
x
x
)(arcsin
1)(
5
1)(
149
23
151
223
423
21
51
32
223
4
21
1049
4
21
10
4
5
5
5
Example (2)
),(
),1()(arccos
1)(
5
1)(
149
532
23
23
151
223
423
21
51
32
223
4
21
1049
4
21
10
4
55
5
5
xlyequivalentor
ch
dxx
dxx
x
dxx
x
dxx
xx
x
x
Example (3)
)3
2ln(,1;
)(arctan6
1
)(13
2
4
1
)(14
1
14
1
94
23
23
223
23
223
249
2
xlyequivalentorewhere
ceh
e
dxe
e
dxe
e
dxe
e
dxe
x
x
x
x
x
x
x
x
x
x
Example (4)
{),(;)(arctan
),(),(;)coth(
223
523
41
51
32
223
5
41
1049
5
41
10
5
5325
32
2
53301
5325
32
2
53301
5
5
)(1
5
)(1
194
xch
Uxcarc
x
x
x
x
dxx
dxx
x
dxx
x
dxx
Logarithmic Expressions of inverse hyperbolic Functions
1;)1ln(arccos.1 2 xxxhx
)1ln(arcsin.2 2 xxhx
)1,1(;)1
1ln(2
1arctan.3
xx
xhx
),1()1,(;)1
1ln(2
1coth.5
Uxx
xxarc
]1,0(;)11
ln(sec.42
xx
xhxarc
Proofs
1;)1ln(
1
12
442
012
21
2
2
cosh
01;arccos
:.1
2
2
22
2
2
xxxy
xxechooseWe
xxxx
e
xee
xee
xee
xee
xy
yandxhxy
Let
y
y
yy
yy
yy
yy
)1ln(
1,
12
442
012
21
2
2
sinh
arcsin
:.2
2
2
22
2
2
xxy
xxethenpositiveallwaysiseSince
xxxx
e
xee
xee
xee
xee
xy
hxy
Let
yy
y
yy
yy
yy
yy
)1,1(;)1
1ln(2
1
)1
1ln(2
1
1
1)1(
)1(1
1
1
tanh
)1,1(;arctan
:.3
2
2
222
2
2
xx
xy
x
xy
x
xe
xex
xxeexe
xe
e
xee
ee
xy
xhxy
Let
y
y
yyy
y
y
yy
yy
]1,0(;)11
ln(
)11
ln(
11,
11
2
442
02
)1(2
1
2
2
sec
]1,0(;sec
:.4
2
2
2
22
2
22
2
xx
xy
x
xy
x
xeChoose
x
x
x
xe
xexe
xxeexe
xe
e
xee
xhy
xhxarcy
Let
y
y
yy
yyy
y
y
yy
),1()1,(;)1
1ln(2
1
)1
1ln(2
1
1
)1(1
)1(1
1
1
coth
),1()1,(;coth
:.5
2
2
222
2
2
Uxx
xy
x
xy
x
xe
exx
xxeexe
xe
e
xee
ee
xy
Uxxarcy
Let
y
y
yyy
y
y
yy
yy
Deducing the derivative formulas for inverse hyperbolic functions using their logarithmic
expressions Question:
Use their logarithmic expressions for the inverse hyperbolic sine, the inverse hyperbolic cosine and the inverse hyperbolic tangent to deduce their derivative formulas
1
11
1
1
1
)1
1(1
1
]2)1(1[1
1
])1(ln[
)1ln(arcsin
)1(
2
2
2
2
22
221
2
2
2
21
21
x
x
xx
xx
x
x
xx
xxxx
y
xx
xxhxy
1;1
11
1
1
1
)1
1(1
1
]2)1(1[1
1
])1(ln[
1;)1ln(arccos
)2(
2
2
2
2
22
221
2
2
2
21
21
xx
x
xx
xx
x
x
xx
xxxx
y
xx
xxxhxy
)1,1(;1
11
1.
1
1
)1(
2.
1
1
2
1
)1(
)1)(1()1(.
1
1
2
1
)1,1(;)1
1ln(2
1arctan
)3(
2
2
2
xx
xx
xx
x
x
xx
x
xy
xx
xhxy
Values
)1,1(2,)2(arctan.2
),1[2,)2(arccos.1
fdombecauseexistnotdoesh
fdombecauseexistnotdoesh
)2
3ln(2
1)
1
1ln(2
1)5
1(arctan.5
)103ln()133ln()3(arcsin.4
)83ln()133ln()3(arccos.3
51
51
2
2
h
h
h