7/31/2019 Lesson 12_Derivative of Inverse Trigonometric Functions

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DIFFERENTIATION OF

INVERSE TRIGONOMETRICFUNCTIONS

7/31/2019 Lesson 12_Derivative of Inverse Trigonometric Functions

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OBJECTIVES:

derive the formula for the derivatives of the

inverse trigonometric functions;

apply the derivative formulas to solve for thederivatives of inverse trigonometric

functions; and

solve problems involving derivatives of

inverse trigonometric functions.

7/31/2019 Lesson 12_Derivative of Inverse Trigonometric Functions

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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.

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The INVERSE TRIGONOMETRIC FUNCTIONS.

x.issinewhoseangletheisymeanalsoThis

xsinyorxarcsinybydenoted

xoffunctionsineinversethecalledisyxysin

relationthebydeterminedxoffunctionaisyif

FunctionsricTrigonometInverseofPropertiesandsDefinition

callRe

1-

-1xif0y2

-or

1xif/2y0:wherexycscifx1cscy

-1xify/2or

1xif/2y0:wherexysecifx1-secy

y0:wherexycotifx1coty

/2y/2-:wherexytanifx1tany

y0:wherexcos yifx1cosy

/2y/2-:wherexysinifx1siny

:sdefinitionfollowingthearethesegeneral,In

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DIFFERENTIATION FORMULADerivative of Inverse Trigonometric Function

functions.rictrigonomet

othertheforformulasthederivecanwemannersimilarIn

x-1

1

dx

xsindxsinybut

x-1

1

dx

dy

x-1ysin-1ycos:identitythefrom

ycos

1

dx

dyor

dy

dxycos

:ytorespectwithtingifferentiaD

2y

2-wherexysinfunction

rictrigonometinverseofdefinitiontheusewe,xsinyofderivativethefindingIn

2

1-1-

2

22

-1

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DIFFERENTIATION FORMULA

Derivative of Inverse Trigonometric Function

dx

du

1uu

1ucsc

dx

d6.

dxdu

1uu1usec

dxd5.

dx

du

u1

1ucot

dx

d4.

dx

du

u1

1utan

dx

d3.

dx

du

u1

1ucos

dx

d2.

dx

du

u1

1usin

dx

d1.

:functionsrictrigonometinverseforformulasationDif ferenti

2

1

2

1

2

1

2

1

2

1

2

1

7/31/2019 Lesson 12_Derivative of Inverse Trigonometric Functions

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A. Find the derivative of each of the following

functions and simplify the result:

31xsinxf.1

223

3x

x1

1(x)f'

6

6

6

2

x1

x1

x1

3xxf'

x3cosxf.2 1

2

2

2 9x19x1

9x13xf'

EXAMPLE:

3

3x1

1xf'

2

6

62

x1

x13xxf'

2

2

9x1

9x13xf'

6

2

x1

3xxf'

29x1

3xf'

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21 x2secy.3

4x12x2x

1

y' 222

14xx

2y'

4

xcos2y.4 1

x2

1

x1

12'y 2

x'y

1x

1

xx1

1

14x14x

14xx2y'

4

4

4

14xx14x2

y'4

4

x-1x

x-1x

x-1x

1

'y

x-1xx-1x'y

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x1 e2sin2

1xh.5

2x

x

e21

e2

2

1x'h

x2

x2

x2

x

e41

e41

e41

e

t5csct5sectg.6 11

5125t5t

1)(5

125t5t

1tg' 22

x2

otcxg.7

1

22x

2

x

21

1x'g

2

2x

x

41

2

4x

2x'g

2

x2

x2x

e41

e41e

0tg'

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x3tanxxf.8 12

x2x3tan3

x31

1xxf

1

2

2

x3tan2

x91

x3xxf 1

2

)x

5(cscSecy.9 1

1x

5csc

x

5csc

x

5

x

5cot

x

5csc

'y 2

2

x

5cot

x

5cot1

x

5csc,but

22

2

'

x

5y

2

12

x91

x3tanx912x3xxf

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A. Find the derivative and simplify the result.

x3tan3xg.1 1

xcot2

1x2sinxy.2

11

3

1

x

4sinxf.3

4x2cscarcy.4

x2Cosx5xG.5 12

xsincosy.6 1

x9

x3cotxF.7

21

x3tansiny.811

211 xsecx6x3sinxh.9

21

5

x5cotx7y.10

EXERCISES:

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x2cos7y.4 1

2

tarcsin4t4ttg.1

2

21xcosy.2

z3secarczzf.3 4

x71tany.5 1

55 yarccosyyh.6

4x

xarcsiny.7

2

y1

y1arctanyF.8

x4cosx4tany.9 11

x4tan

4xxH.10

1

B. Find the derivative and simplify the result.