Download Chapter 3.8

AP CALCULUS AB
Chapter 3:
Derivatives
Section 3.8:
Derivatives of Inverse Trigonometric
Functions
What you’ll learn about
Derivatives
 Derivatives
 Derivatives
 Derivatives
 Derivatives

of
of
of
of
of
Inverse Functions
the Arcsine
the Arctangent
the Arcsecant
the Other Three
… and why
The relationship between the graph of a function
and its inverse allows us to see the relationship
between their derivatives.
Derivatives of Inverse Functions
dy
If f is differentiable at every point of an interval I and
dx
is never zero on I , then f has an inverse and f 1 is differentiable
at every point on the interval f  I  .
Derivative of the Arcsine
If u is a differentiable function of x with u  1, we apply the
Chain Rule to get
d
1 du
sin 1 u 
, u  1.
2
dx
1  u dx
Example Derivative of the Arcsine
If y  sin 1 8 x 2 , find
y  sin 1 8 x 2
where u  8 x 2
dy
1

2
dx
2
1  8x 


1
1  64 x
16 x
dy
.
dx
4
1  64 x 4
 du
2 
8
x




dx


16 x 
You try: Find the derivative of y
1.
2.
y  sin 1 x
y  t sin 1 t 2
Derivative of the Arctangent
The derivative is defined for all real numbers.
If u is a differentiable function of x, we apply the
Chain Rule to get
d
1 du
tan 1 u 
.
2
dx
1  u dx
Derivative of the Arcsecant
If u is a differentiable function of x with u  1, we have the
formula
d
1
du
sec 1 u 
, u  1.
2
dx
u u  1 dx
Example Derivative of the
Arcsecant
Given y  sec 1  3x  4  , find
dy
.
dx
y  sec 1  3 x  4  where u  3 x  4
dy

dx 3 x  4

1
d
 3x  4 
2
 3x  4   1 dx
3
3 x  4 9 x 2  24 x  15
You try:

xt 
defines the position of an object moving along the xaxis at any time t  0.
Find the velocity of the object at the indicated value of t.
1. xt   tan 1 3t
 
2
t
2. xt   sec
4
1
at t  3
at t  3
Inverse Function – Inverse
Cofunction Identities
cos 1 x 
1
cot x 
csc 1 x 

2

2

2
 sin 1 x
 tan 1 x
 sec 1 x
Calculator Conversion Identities
1
sec 1 x  cos 1  
x
cot 1 x 

2
 tan 1 x
1
csc x  sin  
x
1
1
Example Derivative of the
Arccotangent
Find the derivative of y  cot 1 x 2 .
Rewrite y  cot 1 x 2 as y 
dy
1
du 2
0 
x 

2
2
dx
1   x  dx

1
2x
2
x


 
1  x4
1  x4

2
 tan 1 x 2 where u  x 2 .
In Review:

Derivatives of inverse trig functions:
1.
d
1
du
1
sin u 
 ,
dx
1  u 2 dx
2.
d
1 du
1
tan u 

2
dx
1  u dx
3.
d
1
du
1
sec u 

2
dx
u u  1 dx
u 1
In Review:

For the other 3 inverse trig functions, use:
1
cos x 
1
cot x 
1
csc x 

2

2

2
 sin 1 x
 tan 1 x
 sec 1 x
So their derivatives will just be the negatives of
their co-function’s derivative
Section 3.8 – Derivatives of Inverse
Trigonometric Functions

Remember:
1
sec x  cos  
 x
1
1  1 
cot x  tan  
 x
1
1  1 
csc x  sin  
 x
1
1
Section 3.8 – Derivatives of Inverse
Trigonometric Functions
Remember: A function g is an inverse of
the function f if
f(g(x))=x for all x in the domain of g
and
g(f(x))=x for all x in the domain of f.
 The notation for the inverse function of f is
f 1 (reads “f inverse”).
 A function has an inverse function if and
only if it is one-to-one.

Section 3.8 – Derivatives of Inverse
Trigonometric Functions
f and f 1 are reflections
Remember:
across the y = x line, so if (a, b) is in f,
1
f
.
then (b, a) is in
 Remember: Use the Horizontal Line Test
to determine if a function has an inverse
function.

No inverse function
Has an inverse function
Section 3.8 – Derivatives of Inverse
Trigonometric Functions
To find the inverse of a function:
1. Determine if the function has an inverse
function.
2. Interchange x and y.
3. Solve for y. The resulting equation is
y  f 1 x .
4.
5.
1
Define the domain of f
to be the
range of f.
1
1
Verify that f f x   x and f  f x   x.


Section 3.8 – Derivatives of Inverse
Trigonometric Functions
1
f
1. If f is continuous on its domain, then
is continuous on its domain.
1
2. If f is increasing on its domain, then f
is increasing on its domain.
1
3. If f is decreasing on its domain, then f
is decreasing on its domain.
4. If f is differentiable at c and f  c  0
1
then f
is differentiable at f c .

Section 3.8 – Derivatives of Inverse
Trigonometric Functions

If f is differentiable and possesses an
inverse function g, then g is differentiable
at any x for which f g x   0
and
1
g x  
for f g x   0.
f g x 
Section 3.8 – Derivatives of Inverse
Trigonometric Functions
Ex :
f x   2 x  x  1,
a  2
4
2

f x   10 x  3x
f  1  2  a
5

f 
1
3
1
 2 

1
f  f  2
f  1
1
1


4
2
10 1  3 1 13

1
