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