DIFFERENTIATION OF INVERSE TRIGONOMETRIC

FUNCTIONS

OBJECTIVES:• derive the formula for the derivatives of the

inverse trigonometric functions;• apply the derivative formulas to solve for the

derivatives of inverse trigonometric functions; and

• solve problems involving derivatives of inverse trigonometric functions.

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.

The INVERSE TRIGONOMETRIC FUNCTIONS.

x. is sine whoseangle the isy mean also This

xsiny or x arcsin yby denoted

x of function sineinverse the called isy x y sin

relation theby determined x of function a isy if

Functions ric Trigonomet Inverse of Properties and s Definition

callRe

1-

•==

→=•

-1x if 0 y2π- or

1x if π/2 y0 :where x ycsc if x1cscy

-1x if yπ/2 or 1x if π/2 y0 :where x ysec if x1-secy

πy0 :where x ycot if x1coty

π/2yπ/2- :where x ytan if x1tany

πy0 :where x cos y if x1cosy

π/2yπ/2 - :where x ysin if x1siny

:sdefinition following the are these general, In

≤<≤

≥≤<===>−=

≤≤<≥<≤===>=

<<===>−=

<<===>−=

≤≤===>−=

≤≤===>−=

π

DIFFERENTIATION FORMULADerivative of Inverse Trigonometric Function

( )

( )

functions. ric trigonomet

other the for formulas the derive can wemanner similarIn

x-1

1

dx

xsindxsiny but

x-1

1

dx

dy

x-1ysin-1y cos :identity the from

y cos

1

dx

dy or

dy

dx ycos

:y to respect withting ifferentiaD2

y2

- wherexy sin function

ric trigonomet inverse of definition the use we,xsiny of derivative the finding In

2

1-1-

2

22

-1

dxdu

u-11usin

dxd Therefore

21- =

=→==

==

==

≤≤=→

=ππ

DIFFERENTIATION FORMULADerivative of Inverse Trigonometric Function

( )

( )

( )

( )

( )

( )dx

du

1uu

1ucsc

dx

d 6.

dx

du

1uu

1usec

dx

d 5.

dx

du

u1

1ucot

dx

d 4.

dx

du

u1

1utan

dx

d 3.

dx

du

u1

1ucos

dx

d 2.

dx

du

u1

1usin

dx

d 1.

:functions ric trigonomet inverse for formulas ation Differenti

2

1

2

1

21

21

2

1

2

1

−−=

−=

+−=

+=

−−=

−=

−

−

−

−

−

−

A. Find the derivative of each of the following functions and simplify the result:

( ) 31 xsinxf .1 −=

( )( )2

233x

x1

1(x)f'−

=

( )6

6

6

2

x1x1

x13xxf'

−−•

−=

( ) ( )x3cosxf .2 1−=

( )2

2

2 9x19x1

9x13xf'

−−•

−−=

EXAMPLE:

( )( )

( )33x11xf'

2−−=

( ) 6

62

x1x13xxf'

−−= ( ) 2

2

9x19x13xf'

−−−=

( )6

2

x13xxf'

−= ( )

29x13xf'

−−=

( )21 x2secy .3 −=

( )( )4x12x2x

1y'222 −

=

14xx2y'4 −

=

xcos2y .4 1−=

( )

⋅

−

−⋅=x2

1

x1

12'y

2

( )x'y

−−=

⋅−−=

1x1

xx11

14x14x

14xx2y'

4

4

4 −−•

−=

( )14xx14x2y' 4

4

−−=

( )( )( )x-1xx-1x

x-1x1 •−='y

( )( )x-1x

x-1x−='y

( ) ( )x1 e2sin2

1xh .5 −=

( )( ) 2x

x

e21

e2

2

1x'h

−⋅=

x2

x2

x2

x

e41

e41

e41

e

−−•

−=

( ) t5csct5sec tg .6 11 −− +=

( ) ( ) ( )5125t5t

1)(5125t5t

1tg'22 −

−+−

=

( )x2

otc xg .7 1−=

( )

−

+

−=22 x

2

x

21

1x'g

22 x

x4

1

2

⋅

+

= ( )4x

2x'g

2 +=→

x2

x2x

e41

e41e

−−=

( ) 0tg' =

( ) ( )x3tanxxf .8 12 −=

( ) ( ) x2x3tan3x31

1xxf 1

22 •+

•

+= −

( )

+

+= − x3tan2

x91

x3xxf 1

2

)x

5(cscSecy .9 1−=

1x5

cscx5

csc

x5

x5

cotx5

csc'y

2

2

−

−−

=x

5cot

x

5cot1

x

5csc ,but

22

=

=−

2'

x

5y =

( ) ( )( )

+

++=−

2

12

x91

x3tanx912x3xxf

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 .8 11 −−=

( ) 211 xsecx6x3sinxh .9 −− −=

21

5

x5cot

x7y .10 −=

EXERCISES:

( )x2cos7y .4 1−=

( )2

tarcsin4t4ttg .1 2 +−=

21 xcosy .2 −−=

( ) z3secarczzf .3 4=

( )x71tany .5 1 −= −

( ) ( ) 55 yarccosyyh .6 =

+=

4x

x arcsiny .7

2

( )

−+=

y1

y1 arctanyF .8

x4cosx4tany .9 11 −− +=

( )x4tan

4xxH .10

1−

+=

B. Find the derivative and simplify the result.