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

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)

Derivative of an inverse function

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 )

Example:

Derivatives of inverse Functions 𝑑 𝑑𝑥 𝑠𝑖𝑛 −1 𝑢= 1 1 − 𝑢 2 𝑑𝑢 𝑑𝑥 Example: Find the derivative of sin-1(4x3).

Derivatives of inverse tan and sec 𝑑 𝑑𝑥 𝑡𝑎𝑛 −1 𝑢= 1 1+ 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)

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.

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

AP Multiple Choice Practice

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).