DIFFERENTIATION OF INVERSE TRIGONOMETRIC

FUNCTIONS

TRANSCENDENTAL FUNCTIONS

Kinds of transcendental functions:1.logarithmic and exponential functions2.trigonometric and inverse trigonometric functions3.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

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-1

1usin

dxd

Therefore2

1-

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

x1

x1

x1

3xxf'

x3cosxf .2 1

2

2

2 9x1

9x1

9x1

3xf'

EXAMPLE:

33x1

1xf'

2

6

62

x1x13x

xf'

2

2

9x19x13

xf'

6

2

x1

3xxf'

29x1

3xf'

21 x2secy .3

4x

12x2x

1y'

222

14xx

2y'

4

xcos2y .4 1

x2

1

x1

12'y

2

x'y

1x

1

xx1

1

14x

14x

14xx

2y'

4

4

4

14xx14x2

y' 4

4

x-1x

x-1x

x-1x

1

'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)(5

125t5t

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.