Post on 23-Feb-2016
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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