lesson 12 derivative of inverse trigonometric functions
TRANSCRIPT
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.