1. Inverse trigonometric functions and their derivatives
Section 3.8
Inverse trigonometric functions and their derivatives

2. Do-Now: Quiz 1. Find the slope of f(x) at x = 1. f(x) = (x3 – 2)4
2. Find the slope of g(x) at x = π/6
g(x) = cos (5x)

3. Derivative of an inverse function

4. Other notation for derivative of an inverse
If f and g are inverse functions, then
𝑔 ′ 𝑓 𝑥 = 1 𝑓′(𝑥) OR
𝑔 ′ 𝑥 = 1 𝑓′(𝑔 𝑥 )

5. Example:

6. Derivatives of inverse Functions
𝑑 𝑑𝑥 𝑠𝑖𝑛 −1 𝑢= − 𝑢 𝑑𝑢 𝑑𝑥
Example: Find the derivative of sin-1(4x3).

7. Derivatives of inverse tan and sec
𝑑 𝑑𝑥 𝑡𝑎𝑛 −1 𝑢= 𝑢 2 𝑑𝑢 𝑑𝑥
𝑑 𝑑𝑥 𝑠𝑒𝑐 −1 𝑢= 1 𝑢 𝑢 2 −1 𝑑𝑢 𝑑𝑥
Example: Find the derivative of x tan-1(x + 1)

8. Derivatives of the other 3
What is the relationship between sin-1x and cos-1x?
Inverse function-cofunction identities:
𝑐𝑜𝑠 −1 𝑥= 𝜋 2 − 𝑠𝑖𝑛 −1 𝑥
𝑐𝑜𝑡 −1 𝑥= 𝜋 2 − 𝑡𝑎𝑛 −1 𝑥
𝑐𝑠𝑐 −1 𝑥= 𝜋 2 − 𝑠𝑒𝑐 −1 𝑥
Because π/2 is a constant, the derivatives of these three inverse functions will be the negative of their corresponding cofunction.

9. Example
Find the slope of the curve y = csc-1(2x) at x = 3.

10. AP Multiple Choice Practice

11. What do I do if I forget the derivative formulas?
What can you tell me about inverse functions?
1. Switch x and y values.
2. Their graphs are reflections over the line y = x.
3. The slope of f at x is the reciprocal of the slope of f-1 at f(x).