section 6-d inverse trig derivatives. inverse trig functions
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SECTION 6-DInverse Trig Derivatives
Inverse Trig functions
Inverse Trig functions Principal Values
0
2
2
A geometric representation of the range valuesfor each inverse trig function is shown in the coordinate plane at right. arcsin
arctan
arccos
xx 1sin and arcsin
Notation:
1) Evaluate
2
1arcsin )a 0arccos )b 3arctan )c
2) If find tan y
2
5secarcy
A. Find: derive the formula
sin y x
cos 1dyydx
1
cos
dy
dx y
Use implicit differentiation
)sin(sin)sin( 1 xy
)(arcsin xdx
d
)(arcsin xy
A) cont:
1
cos
dy
dx y
2 2sin cos 1y y 2 2cos 1 siny y
2cos 1 siny y
But2 2
y
so is positive.cos y
2cos 1 siny y
2
1
1 sin
dy
dx y
2
1
1
dy
dx x
sin y x
)(arcsin xdx
d
B) find
1tand
xdx
Derivatives of Inverse Trig functions
2
1
1sin
u
uu
dx
d
2
1
1cos
u
uu
dx
d
21
1tan
u
uu
dx
d
1sec
2
1
uu
uu
dx
d
21
1cot
u
uu
dx
d
1csc
2
1
uu
uu
dx
d
3) Find the derivative of )arcsin( 2xy
4) Find the derivative of )5arctan( xy
5) Find the derivative of
1)(
12222
xxxx
xy
)(sec 21 xxy
6) Find the derivative of )arctan( xey
7) Find the derivative of
)1()(sin1
1sin2' 21
2
1
x
xxxy
PRODUCT RULE!
21 )(sin xxy
8) Find the derivative of
12
22
x
xy
1
12
xx
1
1
1 22
2
xxxx
xy
1
12
2
xx
xy
)(sec 1 12 xxy
)(sec 1)-( 1212 xxy
9) Find the derivative of
22
2
11
1
x
x
x
y1
12
xx
1
1
1 222
xxxx
xy
1
22
xx
y
xxy cscarc 1arctan 2
1
1
1
122
xxxx
y
Homework
Page 379 # 3-19, 27,29, 43-48, 50, 51, 53, 54, 57, 58, 59 and 61
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