1. Warm Up: pg 164 #59-61, 63, 64 and pg 156 QQ #4
61) A

2. Derivatives of Inverse Trig Functions
Ch 3.8
Derivatives of Inverse Trig Functions

3. Review from PreCalc: sin(θ) = 𝑜𝑝𝑝 ℎ𝑦𝑝
a ratio
an angle
60° 2 1
EX: sin(30°) = 1 2
30° 3
But inverse trig asks the opposite question
EX: Arcsin( 1 2 ) = find the angle whose ratio is 1 2
=30°

4. The 3 main Inverse Trig Functions

5. Remember the restricted ranges …
Sine and Tangent are in Quadrant I and IV
Cosine is in Quadrant I and II

6. The Reciprocal Function is NOT the inverse function
Inverse notation: Reciprocal Notation: arcsin(ratio) or 𝑠𝑖𝑛 −1 (𝑟𝑎𝑡𝑖𝑜) 1 sin⁡(30°) = csc(30°)
This does not mean reciprocal

7. What is the derivative of the arcsine?
** YOUR FRQ on the next TEST will include deriving one of the 3 main trig functions
We find the derivative of y = arcsin(x) as follows:
Start with y = sin(x)
Inverse: x = sin(y)
Take d/dx = cos(y)y’
Isolate y’ cos⁡(𝑦) =𝑦′
Replace cos(y)
With info from − 𝑥 2 =𝑦′
Triangle picture 1 x y
1− 𝑥 2
We know y = arcsin(x) only exists between - π 2 ≤ y ≤ π 2 and -1 < x < 1

8. What is the derivative of the arctan?
** YOUR FRQ on the next TEST will include deriving one of the 3 main trig functions
Start with y = tan(x)
Inverse: x = tan(y)
Take d/dx = sec²(y)y’
Isolate y’ sec²(𝑦) =𝑦′
Replace cos(y)
With info from 𝑥² =𝑦′
Triangle picture
1− 𝑥 2 x 1

9. You Try! y = arccos(x) Start with y = cos(x) Inverse: x = cos(y)
Take d/dx = -sin(y)y’
Isolate y’ sin(𝑦) =𝑦′
Replace cos(y)
With info from − 𝑥 2 =𝑦′
Triangle picture 1
1− 𝑥 2 x

10. What are the formal definitions of the inverse trig functions”
If u is a differentiable function of x with |u| < 1, we apply the chain rule to get:
𝑑 𝑑𝑥 𝑠𝑖𝑛 −1 𝑢= 1 1−𝑢² 𝑑𝑢 𝑑𝑥 𝑑 𝑑𝑥 𝑐𝑜𝑠 −1 𝑢=− 1 1−𝑢² 𝑑𝑢 𝑑𝑥
𝑑 𝑑𝑥 𝑡𝑎𝑛 −1 𝑢= 1 1+𝑢² 𝑑𝑢 𝑑𝑥 𝑑 𝑑𝑥 𝑐𝑜𝑡 −1 𝑢=− 1 1+𝑢² 𝑑𝑢 𝑑𝑥
𝑑 𝑑𝑥 𝑠𝑒𝑐 −1 𝑢= 1 |𝑢| 𝑢 2 −1 𝑑𝑢 𝑑𝑥 𝑑 𝑑𝑥 𝑐𝑠𝑐 −1 𝑢=− 1 |𝑢| 𝑢 2 −1 𝑑𝑢 𝑑𝑥

11. Find Full Assignment Online

12. EX 1: Applying the Arcsine formula
a) Find 𝑑 𝑑𝑥 ( 𝑠𝑖𝑛 −1 ( 3 𝑥² )) b) Find 𝑑 𝑑𝑥 ( 1 𝑠𝑖𝑛 −1 (2𝑥) )

13. EX 2: Applying the Arctan formula
A particle moves along the x-axis so its position at any time t ≥ 0 is x(t) = 𝑡𝑎𝑛 −1 ( 𝑡 ). What is the velocity of the particle when t = 16?

14. You try!
𝑑 𝑑𝑥 ( 𝑠𝑖𝑛 −1 (1−𝑥)) b) Find 𝑑 𝑑𝑥 ( 𝑐𝑠𝑐 −1 ( 𝑥 2 )) c) 𝑑 𝑑𝑥 ( 𝑠𝑒𝑐 −1 (5 𝑥 4 ))
d) A particle moves along the x-axis so its position at any time t ≥ is
x(t) = 𝑠𝑖𝑛 −1 ( 𝑡 4 ). What is the velocity of the particle when t = 4?

15. EX 3: Derivatives of inverse functions
a) Find an equation for the line tangent to the graph of y = tan(x) at the point ( 𝜋 4 , 1)
b) Find an equation for the line tangent to the graph of y = 𝑡𝑎𝑛 −1 (x) at the point (1, 𝜋 4 )

16. What did you notice in the last EX?
*The Derivative of a function is the reciprocal of the derivative of its inverse*

17. EX 4: Derivatives of inverse functions
Let f(x) = 𝑥 5 +2 𝑥 3 +𝑥−1.
Find f(1) b) Find f’(1)
c) Find 𝑓 −1 (3) d) Find 𝑓 −1 (3)′
So for #28:
F(1) = 3 f’(1) =12
(3,1) f’(3) = 1/12

18. You Try! Let f(x) = cos 𝑥 +3𝑥 Find f(0) b) Find f’(0)
c) Find 𝑓 −1 (1) d) Find 𝑓 −1 (1)′

19. 3.8: pg 170 QR #1-3, 6- 10, EX: #1- 19odd, 24, 26, 29
DIDN’T GET TO 3.8B