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Logarithmic, Exponential, and
Other Transcendental Functions5
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I you aren!t in over yourhead, ho" do you #no"
ho" tall you are$
T. %. Eliot
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Inverse Trigonometric
Functions& 'ierentiation
Copyright © Cengage Learning. All rights reserved.
5.6
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(
)nder suita*le restrictions, each o the six trigonometricunctions is one+to+one and so has an inverse unction, as
sho"n in the ollo"ing deinition.
Inverse Trigonometric Functions
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-
The graphs o the six inverse trigonometric unctions aresho"n in Figure ./(.
Figure 5.29
Inverse Trigonometric Functions
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Example 0 Evaluating Inverse Trigonometric Functions
Evaluate each unction.
%olution&
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/
cont1dExample 0 Evaluating Inverse Trigonometric Functions
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2
Inverse unctions have the properties
f 3f 0,3 x 44 5 x and f 0,3f 3 x 44 5 x .
6hen applying these properties to inverse trigonometric
unctions, remem*er that the trigonometric unctions have
inverse unctions only in restricted domains.
For x +values outside these domains, these t"o properties
do not hold.
For example, arcsin3sin π 4 is e7ual to -, not π .
Inverse Trigonometric Functions
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Inverse Trigonometric Functions
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Example / 0 Solving an Equation
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9
Inverse Trig Functions
Given arcsin , where 0 / 2, find cos . y x y yπ = < <sin y x=
y
1 x
2
1 x−2cos 1 y x= −
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:
Inverse Trig Functions
( )5 /Gi 2ven arcsec , find tan . y y=
y
5
2
tan 1/ 2 y =
5sec
2
y = 1
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;
Inverse Trig Functions
1sec tan3
x Find −
÷
y x
3tan3
x y =
21sec tan
3
9
3
x x− = ÷
+
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(
'erivatives o Inverse Trigonometric Functions
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/-
Inverse Trig Functions and 'ierentiation
1sin , y u Find y− ′=
yu
1sin y u=
21sec tan
3
9
3
x x− = ÷
+
Ta#e derivative implicitly
( )cos y y u′ ′=
( )cosu y
y′′ =
?
21
u y
u
′′ =−
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Inverse Trig Functions and 'ierentiation
1tan , y u Find y− ′=
yu
1
tan y u=
21sec tan
3
9
3
x x− = ÷
+
Ta#e derivative implicitly
( )2sec y y u′ ′=
( )22
2sec 1
u u y
yu
′ ′′ = =+
?
2 1
u y
u
′′ =
+
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//
The ollo"ing theorem lists the derivatives o the six inversetrigonometric unctions.
'erivatives o Inverse Trigonometric Functions
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Inverse Trig Functions and 'ierentiation
[ ]arcsin 2d
xdx
=
[ ]2
'arcsin1
d uudx u
=−
2
2
1 4 x−
[ ]arctan 3d
xdx =
[ ]2
'arctan
1
d uu
dx u
=
+
2
3
1 9 x+
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'erivatives o Inverse Trigonometric Functions
arcsind
xdx
=
1
2
/ 2 1
1 2 1
1/
2
1 2 x
x x x x x
−
= =− −
−
[ ]2
'arcsin
1
d uu
dx u=
−
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/9
2 4
2
2 2( )
2
1
2
1
x
x x xe e e
e
= −−
'erivatives o Inverse Trigonometric Functions
2arcsec xd
edx
= [ ] 2'
arcsec1
d uu
dx u u=
−
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'erivatives o Inverse Trigonometric Functions
2 arcsin 1 Differentiate y x x x= + −
[ ]2
'arcsin
1
d uu
dx u
=
−
( ) ( )
( )
1/ 22
22 2
2 2
2
2
2
2 1' 1 22
2 112
1
1
11
1
1
1 1
y x x x
x
x
x x x
x x
x
x
− = + − − ÷
−+ −= − = =
− − −
−
−
+
−
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/(
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S!!ar" of #ifferentiation $%es
[ ]
[ ]
[ ]
[ ]
[ ]
( )
2
1
1. '
2. ' '
3. ' '
' '4.
5. 0
&. '
. 1
(. ' , 0
n n
d cu cu
dx
d u v u vdx
d uv u v uv
dx
d u u v uv
dx v v
d c
dx
d u nu u
dx
d x
dx
d uu u u
dx u
−
=
± = ±
= +
− =
=
=
=
= ≠
[ ]
[ ]( )
( )
[ ] ( )
[ ] ( )
[ ] ( )
[ ] ( )
2
2
'9. %n
10. '
'11. %o)
%n
12. %n '
13. sin cos '
14. cos sin '
15. tan sec '
1&. cot csc '
u u
a
u u
d uu
dx u
d e e udx
d uu
dx a u
d
a a a udx
d u u u
dx
d u u u
dxd
u u udx
d u u u
dx
=
=
=
=
=
= −
=
= −
[ ] ( )
[ ] ( )
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
2
2
2
2
2
2
1. sec sec tan '
1(. csc csc cot '
'19. arcsin
1
'20. arccos
1
'21. arctan
1
'22. cot
1
'23. sec
1
'24. csc
1
d u u u u
dx
d u u u udx
d uu
dx u
d uu
dx u
d uu
dx u
d uarc u
dx u
d uarc u
dx u u
d uarc u
dx u u
=
= −
=−−
=−
=+−
=
+=
−
−=
−
coefficient * e+,ressions in ter!s ofc u v x→ →
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2
#a" 1 - 5.& .3
1, 21, 31, 41&1,1, a%% odds
#a" 2 - )s.205,20&
>ome"or#
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29
'ay / ?ore Examples. Diferentiate:
( )5arctan.2 2 −= x y
( )25arcsin.1 += x y
( )12csc.331 +−= − x x y
= −
&cot.4 1
x y
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2:
@ractice @ro*lems. Diferentiate:
( )93arccos4.2 += x y
x x y arccosarcsin(.1 +=
−= −
(tan4.3 1
x y
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2;
Calculus >6 B/8
Find the e7uation o the tangent line to the graph o the
unction at the given point&
arctan 2,2 4
x
y at
π
= ÷ ÷
( )1
24 4
y xπ
− = −