Derivs of Inverse Trig Func 2014

8/18/2019 Derivs of Inverse Trig Func 2014

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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



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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 .



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8



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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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:


( )5 /Gi 2ven arcsec , find tan . y y=

y

5

2

tan 1/ 2 y =

5sec

2

y = 1


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;


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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/


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.



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/8


[ ]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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/


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

= −−


2arcsec xd

edx

= [ ] 2'

arcsec1

d uu

dx u u=

−


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/:


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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22

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

− = −

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Derivs of Inverse Trig Func 2014