Download Derivatives of Inverse Functions

Chapter 5 Derivatives of
Transcendental Functions
5-1 The Natural Logarithmic Function
5-4 Exponential Functions
5-5 Bases other than e and Applications
5-3 Inverse Functions and their Derivatives
5-8 Inverse Trig Functions
7-7 Indeterminate Forms and L’Hopital’s Rule
5.8 Derivatives of Inverse
Trigonometric Functions
We can use implicit
differentiation to find:
y  sin 1 x
sin y  x
dy
cos y
1
dx
dy
1

dx cos y
d
sin 1 x
dx
d
d
sin y 
x
dx
dx
dy
1

dx
1  sin 2 y
dy
1

dx
1  x2
sin 2 y  cos2 y  1
cos2 y  1  sin 2 y
cos y   1  sin 2 y
But 

 y

2
2
so cos y is positive.
 cos y  1  sin 2 y
The derivatives of arcsecx and arctanx are found in a similar manner
using the trig identity sec2 y  1  tan2 y .
Derivatives of Inverse
Trigonometric Functions
d
1
du
1
sin u 
dx
1  u 2 dx
d
1 du
1
tan u 
dx
1  u 2 dx
d
1
du
sec 1 u 
dx
u u 2  1 dx
Derivatives of the other Three
There is a much easier way to find the other three inverse
trigonometric functions using the following identities:
It follows easily that the derivatives of the inverse cofunctions are the negatives of the derivatives of the
corresponding inverse functions.
Derivatives of Inverse
Trigonometric Functions
d
1
du
1
sin u 
dx
1  u 2 dx
d
1 du
1
tan u 
dx
1  u 2 dx
d
1
du
sec 1 u 
dx
u u 2  1 dx
d
1 du
1
cos u  
dx
1  u 2 dx
d
1 du
cot 1 u  
dx
1  u 2 dx
d
1
du
csc 1 u  
dx
u u 2  1 dx
dy
Examples: Find
of the given functions.
dx
1. y  cos (3x )
dy
1
6x

(6 x )  
2
2
dx
(1  (3x )
1  9 x4
1
2. y  cot  
 x
dy
1  1
1

 2   2
1  x  x 1
dx
1 2
x
dy
1
x
 (sec 1 x)(1)
dx
| x | x2  1
1
1
3. y  x sec1 x
2
4. y  arcsin x  x 1  x 2
4. Solution
y  arcsin x  x 1  x
2
1 
2 1 2
2
y'
 x 2x 1 x   1 x
2
2 
1 x
1
1

1 x

2
x


2 1 x
2
1 x
2
2
1 x

 1 x 
2
2
 2 1 x
2
1 x 1 x
2
1 x
2
2
You Try…
Find the derivative of the following functions.
1. f ( x)  arctan x
1
f ( x) 
2 x
1
 x
2

1
2 x 1  x 
2. g ( x)  e  arcsin( x)
x
3. h( x)  arc sec(e )
2x

2e2 x
2x
e
e 
2x 2
1

2
e4 x  1
You Try…
1
1
4. Given f ( x)  cos ( x), find the equation
2

2 3 
of the line tangent to this function at  

2
,

8 
.
Closure
Give the derivatives of all 6 inverse trig
functions and a way for memorizing each.