2016 Derivatives of Inverse Trig Functions

AP Calculus

Inverse Trig Functions

y = sin(x)

sin(y) = x

y = sin-1(x) (with restrictions)

Inverse Trig Functions with Restrictions

Remember : Implicit differentiation

2 2 2 3y x xy

Remember:

Pythagorean Relations

A C

B

a

b

c

a or b unknown c unknown

y

Inverse Trig: Arcsine

1Find if sin ( )dx y xdy

sin( )y x

y

x1

2(1 )x

1

2

1sin ( )(1 )

d xdx x

Chain Rule:

EX:Find the derivative:

1 4( ) sin ( )f x x

Find the derivative.

1( ) sinf x x x

Inverse Trig: Arctangent

tan( )y x

y

x

1

2( 1)x

12

1tan ( )1

d xdx x

Chain Rule:

1Find if tan ( )dx y xdy

EX:Find the derivative:

1( ) arctan1

xf xx

Inverse Trig: Arcsecant

sec( )

(sec( )) ( )

sec( ) tan( ) 1

1sec( ) tan( )

y xd dy xdx dx

dyy ydx

dydx y y

y

x

1

2( 1)x

1

2

1sec( 1)

d xdx x x

Chain Rule:

1Find if sec ( )dx y xdy

Arcsine with the Chain Rule: u = f(x)

1

2

1sin ( )(1 )

d u udx u

Arccosine with the Chain Rule: u = f(x)

1

2

1cos ( )(1 )

d u udx u

Learn these in PAIRS:

Arctangent with the Chain Rule: u = f(x)

12

1tan ( )1

d u udx u

Arccotangent with the Chain Rule: u = f(x)

12

1cot ( )1

d u udx u

Learn these in PAIRS:

Inverse Trig: with the Chain Rule: u = f(x)

1

2

1sin ( )(1 )

d u udx u

1

2

1cos ( )(1 )

d u udx u

Learn these in PAIRS:

12

1tan ( )1

d u udx u

12

1cot ( )1

d u udx u

1

2

1sec( 1)

d u udx u u

1

2

1csc( 1)

d u udx u u

Last Update

• 10/19/10

• Assignment: p 170 # 1- 21 odd , 27-29