Name:
Block:
Date: __________ AP Calculus
Sec 3.8 Derivatives of Inverse Functions and Inverse
Trigonometric Functions
5
Let f (x ) = x + 2x − 1 .
a) Since the point (1,2) is on the graph of f, why does it follow that the point (2,1) is on the
−1
graph of f ?
Ex 1
b) Determine
df −1
(2 )
dx
′
c) Develop a shortcut formula for (f −1 ( x ))
(f
Ex 2 Let f (x ) = x 3 + x 2 + 1
a) Find f (1) and f ′(1)
′
−1
b) Find f (3) and (f −1 ) (3)
−1
′
(x )) =
Derivative of the Arcsine Function
Consider the function y = sin x
What is the inverse of this function:
Solve for y:
Use your graphing calculator to graph this inverse trigonometric function:
What is the domain and range?
π
y
x
Determine the derivative of the inverse sine function:
−π
−1
Two new formulas:
′
′
(Arcsin x ) = (sin −1 x ) =
Ex 3 Determine:
d
−1 2
sin x )
(
dx
′
′
(Arcsin u) = (sin −1 u) =
1
Derivative of the Arctangent Function
Consider the function y = tan x
What is the inverse of this function:
Solve for y:
Use your graphing calculator to graph this inverse trigonometric function:
What is the domain and range?
π
y
x
Determine the derivative of the inverse tangent function:
−π
−1
1
Two new formulas:
′
′
(Arctanx ) = (tan−1 x ) =
′
′
(Arctanu) = (tan−1 u ) =
Ex 4
A particle moves along a line so that its position at any time t ≥ 0 is s (t ) = tan −1 t . What is
the velocity of the particle when t = 16 ?
Ex 5
 π

a) Find an equation for the line tangent to the graph of y = tan x at the point  − ,−1 .
 4 

π
b) Find an equation for the line tangent to the graph of y = tan −1 x at the point  −1,−  .

4
There are other inverse trigonometric functions. Here are their derivative formulas:
(cos u)′ =
−1
(cot u)′ =
(sec u)′ =
(csc u)′ =
−1
−1
Ex 6 Find
−1
dy
1
for y = sin
dx
x
−1
2
Additional Question: Given y = sin x − 1− x , determine y ′ .