1. DIFFERENTIATION OF
HYPERBOLIC FUNCTIONS

2. TRANSCENDENTAL FUNCTIONS
Kinds of transcendental functions:
1.logarithmic and exponential functions
2.trigonometric and inverse trigonometric
functions
3.hyperbolic and inverse hyperbolic functions
Note:
Each pair of functions above is an inverse to
each other.

5. HYPERBOLIC IDENTITIES

6. DIFFERENTIATION FORMULA
Derivative of Hyperbolic Function

7. A. Find the derivative of each of the following functions
and simplify the result:
x2coshxsinhy.1 =
)xcoshxsinh(xcosh'y
)xsinhx(coshxcosh)xcoshxsinh(xsinh'y
xcoshxcoshxsinhxsinh'y
22
22
5
22
222
+=
++=
+=
xhsecxy.2 =
xhsec)xtanhxhsec(x'y +−=
xhsecy.3 2
=
xtanhxhsecxhsec2'y −=
xhsecxcothy.5 =
)xhcsc(xhsec)xtanhxhsec(xcoth'y 2
−+−=
2
xsinhlny.4 =
2
2
xsinh
xcoshx2
'y =
EXAMPLE:
)xtanhx1(xhsec'y −=
xtanhxhsec2'y 2
−=
2
xcothx2'y =
[ ]xhcscxtanhxcothxhsec'y 2
+−=
[ ]xhcsc1xhsec'y 2
+−=
hxcscxcothy
xcothxhsec'y
'
−=
−= 2

8. xcothlny.6 2
=
xcoth
xhcscxcoth2
'y 2
2
−
=
xcotharccosy.7 =
xcoth1
xhcsc
'y
2
2
−
−
−=
xhcscxhcsc
xhcscxhcsc
'y
22
22
−•−
−•
=
xsinh
xcosh
xsinh
1
2
'y
2
−
=
2
2
xsinhxcosh
2
'y •
−
=
x2sinh
4
'y −=
x2hcsc4'y −=
xhcsc
xhcsc
'y
2
2
−
=
xhcsc'y 2
−=

9. )xharctan(siny.8 2
=
( )22
2
xcosh
xcoshx2
'y =
2
xhsecx2'y =
( )22
2
xsinh1
xcoshx2
'y
+
=

10. A. Find the derivative and simplify the result.
( ) 2
xsinhxf.1 =
( ) w4hsecwF.2 2
=
( ) 3
xtanhxG.3 =
( ) 3
tcoshtg.4 =
( )
x
1
cothxh.5 =
( ) ( )xtanhlnxg.6 =
EXERCISES:
( ) ( )ylncothyf.7 =
( ) xcoshexh.8 x
=
( ) ( )x2sinhtanxf.9 1−
=
( ) ( )x
xsinhxg.10 =
( ) ( )21
xtanhsinxg.11 −
=
( ) 0x,xxf.12 xsinh
>=

11. Hyperbolic Functions Trigonometric Functions
1xsinhxcosh 22
=−
xhsecxtanh1 22
=−
xhcsc1xcoth 22
=−
ysinhxcoshycoshxsinh)yxsinh( ±=±
( ) ysinhxsinhycoshxcoshyxcosh ±=±
( )
ytanhxtanh1
ytanhxtanh
yxtanh
±
±
=± ( )
ytanxtan1
ytanxtan
yxtan

±
=±
( ) ysinxsinycosxcosyxcos =±
( ) ysinxcosycosxsinyxsin ±=±
xx 22
sectan1 =+
1sincos 22
=+ xx
xcsc1xcot 22
=+
Identities: Hyperbolic Functions vs. Trigonometric Functions

12. Hyperbolic Functions Trigonometric Functions
Identities: Hyperbolic Functions vs. Trigonometric Functions
sinh 2x = 2 sinh x cosh x
( ) 2/1x2coshxsinh2
−=
( ) 2/1x2coshxcosh2
+=
x
exsinhxcosh =+
x
exsinhxcosh −
=−
( ) 2/x2cos1xcos2
+=
( ) 2/x2cos1xsin2
−=
cos 2x = cos2
x – sin2
x
sin 2x = 2sinx cosx
cosh 2x = cosh2
x +sinh2
x