Lesson 3 derivative of hyperbolic functions

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:
x2 cosh x sinhy .1=
)xcoshxsinh(xcosh'y
)xsinhx(coshxcosh)xcoshxsinh(xsinh'y
x cosh x coshx sinh x sinh'y
22
22
5
22
222
+=
++=
+=

x hsecxy .2=
x hsec)x tanhx hsec(x'y +-=

xhsecy .3
2
=
x tanhx hsecx hsec2'y-=

x hsecx cothy .5=
)xhcsc(x hsec)x tanhx hsec(xcoth'y
2
-+-=

2
x sinhlny .4=
2
2
xsinh
xcosh x2
'y=
EXAMPLE:
)x tanh x1(x hsec'y -=
x tanhxhsec2'y
2
-=
2
xcoth x2'y=
[ ]xhcscx tanhx cothx hsec'y
2
+-=
[ ]xhcsc1x hsec'y
2
+-=
hxcscxcothy
xcothx hsec'y
'
-=
-=
2

8. xcoth lny .6
2
=
xcoth
x hcsc x coth2
'y
2
2
-
=
xcotharccosy .7=
xcoth1
xhcsc
'y
2
2
-
-
-=
xhcscxhcsc
xhcscxhcsc
'y
22
22
-·-
-·
=
x sinh
x cosh
xsinh
1
2
'y
2
-
=
2
2
x sinhxcosh
2
'y ·
-
=
x2 sinh
4
'y-=
x2 hcsc4'y-=
xhcsc
xhcsc
'y
2
2
-
=
xhcsc'y
2
-=

9. )x h arctan(siny .8
2
=
( )
2
2
2
x cosh
x cosh x2
'y=
2
x hsec x2'y=
( )
2
2
2
x sinh1
x cosh x2
'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
( )2/1x2coshxsinh
2
-=
( )2/1x2coshxcosh
2
+=
x
exsinhxcosh =+
x
exsinhxcosh
-
=-
( )2/x2cos1xcos
2
+=
( )2/x2cos1xsin
2
-=
cos 2x = cos
2
x – sin
2
x
sin 2x = 2sinx cosx
cosh 2x = cosh
2
x +sinh
2
x