1. Derivatives of Inverse Trig Functions
Section 3.8
Derivatives of Inverse Trig Functions

2. Theorem: Derivatives of Inverse Functions
If f is differentiable at every point of an interval I and df/dx is never zero on I, then f has an inverse and f-1 is differentiable at every point of the interval f(I).

3. Derivative of the Arcsine
𝑑 𝑑𝑥 𝑠𝑖 𝑛 −1 𝑢= 1 1− 𝑢 2 𝑑𝑢 𝑑𝑥 , 𝑢 <1

4. Example 1
𝑑 𝑑𝑥 𝑠𝑖 𝑛 −1 𝑥 2

5. Example 2
Find dy/dt if y=si 𝑛 − 𝑡 .

6. Derivative of the Arctangent
𝑑 𝑑𝑥 𝑡𝑎 𝑛 −1 𝑢= 1 1+ 𝑢 2 𝑑𝑢 𝑑𝑥

7. Example 3
A particle moves along the x-axis so that its position at any time t ≥ 0 is 𝑥 𝑡 = 𝑡𝑎 𝑛 −1 𝑡 . What is the velocity of the particle when t = 16?

8. Derivative of the Arcsecant
𝑑 𝑑𝑥 𝑠𝑒 𝑐 −1 𝑢= 1 𝑢 𝑢 2 −1 𝑑𝑢 𝑑𝑥 , 𝑢 <1

9. Example 4
Find dy/dx if 𝑦=𝑠𝑒 𝑐 −1 5 𝑥 4 .

10. TOTD
Find the derivative of y with respect to the appropriate variable.
𝑦= sin −1 (1−𝑡)

11. Example 5
Find dy/ds if 𝑦=𝑠𝑒 𝑐 −1 2𝑠+1 .

12. Derivatives of the Other Three
𝑑 𝑑𝑥 𝑐𝑠 𝑐 −1 𝑢=− 1 𝑢 𝑢 2 −1 ∙ 𝑑𝑢 𝑑𝑥
𝑑 𝑑𝑥 𝑐𝑜 𝑠 −1 𝑢=− 1 1− 𝑢 2 ∙ 𝑑𝑢 𝑑𝑥
𝑑 𝑑𝑥 𝑐𝑜 𝑡 −1 𝑢=− 1 1+ 𝑢 2 ∙ 𝑑𝑢 𝑑𝑥

13. Example 6
Find dy/ds if 𝑦=𝑠 1− 𝑠 2 +𝑐𝑜 𝑠 −1 𝑠.

14. Example 7
Find an equation for the line tangent to the graph of 𝑦=𝑐𝑜 𝑡 −1 𝑥 at x = -1.

15. TOTD
Find the derivative of y with respect to the appropriate variable.
𝑦= cot −1 1 𝑥 − tan −1 𝑥