AP Calc ­ Aim #34 ­ derivatives of inverse trig functions.notebook
Unit 4 ‐ Derivatives of Inverses
Aim #34: What are the derivatives of inverse
trigonometric functions?
Do Now: Solve the following problems:
a. arcsin(1)
-1
b. cos (
)
c. arctan (1)
None of the six basic trigonometric functions has an inverse
To compensate for this, we restrict each one to be
one-to-one (pass the horizontal line test)
Function
Domain
Range (quadrant)
y = arcsin x
-1 ≤ x ≤ 1
I and IV
y = arccos x
-1 ≤ x ≤ 1
I and II
y = arctan x
-∞ < x < ∞
I and IV
y = arccsc x
|x|≥1
I and IV (not 0)
y = arcsec x
|x|≥1
I and II (not π/2)
y = arccot x
-∞ < x < ∞
I and II
1
AP Calc ­ Aim #34 ­ derivatives of inverse trig functions.notebook
eg. Evaluate: Give your answer in degrees
a. arcsin
60
b. arccot (1)
o
45
d. arccos (0.94)
19.9
eg. Solve:
c. arctan
o
-30
o
e. arccsc (-1.8)
o
-33.7
o
arccos (2x + 3) =
-1.5
2
AP Calc ­ Aim #34 ­ derivatives of inverse trig functions.notebook
eg. Given y = arcsec (5/3), find cot y.
3/4
Derivatives of inverse trigonometric functions
3
AP Calc ­ Aim #34 ­ derivatives of inverse trig functions.notebook
eg. Find the derivative:
a.
b.
eg. Find the derivative:
a. y = arccos (ln x)
b. y = arctan (5x - 2)
4
AP Calc ­ Aim #34 ­ derivatives of inverse trig functions.notebook
eg. Find the derivative:
x
a. y = arccsc (4 )
2x+1
b. y = arctan (e
)
Wrap it Up!
Homework: Homework #34 - pp 273-275: 1, 5, 23-31 odd
5