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f (x) = y = arcsin x find y’ f (x) = y = arctan x find y’
f (x) = y = arcsec x find y’ Formulas:
2 2 2
2 2 2
2 2 2
d u du uu C
dx a1 u a u
d u du 1 uu C
dx a a1 u a u
ud u du 1u C
dx a au u 1 u u a
'arcsin arcsin
'arctan arctan
'arc sec arc sec
= = +⎡ ⎤⎣ ⎦− −
= = +⎡ ⎤⎣ ⎦ + +
= = +⎡ ⎤⎣ ⎦− −
∫
∫
∫
Give the domain of and compute its derivative.
( ) ( )( )x xf arctan ln=
Give the domain of ( )xe
x2
g arcsin= and find the equation for the
tangent line to the graph of this function at x = 0.
#12. Differentiate: 1y xtan−= #16. Differentiate: ( ) x1x e tanf
−=
#20. Differentiate: 1 2y x 2sec −= +
#40. Evaluate: 1
21
dx
1 x− +∫
#42. Evaluate: 1
0 2
dx
4 x−∫
#46. Evaluate: ( )
5
22
dx
9 x 2+ −∫
4
xdx
1 x−∫
Watch for details. These look similar.
2
1dx
16 x+∫
2
xdx
16 x+∫
Hyperbolic Functions
Section 7.8
Primary Definitions
Hyperbolic cosine: Hyperbolic sine:
( )
( )
x x
x x
e ex
2
e ex
2
cosh
sinh
−
−
+=
−=
( ) ( )( ) ( ) ( )
( )
( ) ( ) ( ) ( )
x xx x
x x
1 1x x
x x
sinh coshtanh coth
cosh sinh
sech cschcosh sinh
= =
= =
Why are these called Hyperbolic Functions? Recall that sine and cosine are called circular functions. Remember the unit circle?
−3 −2 −1 1 2 3
−5
5
10
−3 −2 −1 1 2 3
−5
5
10
( )
( )x
x
x
xex
2
e e
e
x2
s
s
inh
co h−
−
+
−
=
=
( )
( ) x
x
x
x e
eg
f
−=
=
Hyperbolic Identity Show that
( ) ( )2 2t t 1cosh sinh− =
sinh (–x) = – sinh x (odd) cosh (–x) = cosh x (even)
cosh x + sinh x = cosh x – sinh x =
( )dx
dxsinh =
( )dx
dxcosh =
Derivatives and Integral formulas:
2
2
dx x
dx
dx
dx
dx x
dx
dx x x
dx
dx x
dx
dx x x
dx
sinh cosh
cosh sinh
tanh sec h
sec h tanh sec h
coth csc h
csc h coth csc h
=
=
=
= −
= −
= −
1. ( )( )dx
dxtanh ln⎡ ⎤
⎢ ⎥⎣ ⎦
2. ( )( )xd
dx 1 x
cosh
sec h
⎡ ⎤⎢ ⎥
+⎢ ⎥⎣ ⎦
3. ( ) ( )( )x x dxtanh ln cosh =∫
4. ( )2 2x x dxsec h =∫
5. Find the area bounded by the graphs of and for .
( )x xf s inh=( )x xg cosh= 0 x 10ln≤ ≤
1 2 3
5
6.
( )2 3x dxcosh∫
7.7
Differentiate:
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