Chapter 4
Calculating the
Derivative
JMerrill, 2009
Review

Find the derivative of (3x – 2x2)(5 + 4x)
-24x2 + 4x + 15
 Find the derivative of

2

5x  4 x  5
2
(x  1)
2
5x  2
x
2
1
4.3
The Chain Rule
Composition of Functions





A composition of functions is simply putting 2
functions together—one inside the other.
Example: In order to convert Fahrenheit to
Kelvin we have to use a 2-step process by first
converting Fahrenheit to Celsius.
5
C  (F  3 2)
o
o
89 F = 31.7 C
9
K  C  273
31.7oC = 304.7K
But if we put 1 function inside the other function,
then it is a 1-step process.
Composition of Functions
The composite of f(x) and g(x) is denoted  f
which means the same as f(g(x)).
g
x
We are used to writing f(x). f(g(x)) simply
means that g(x) is our new x in the f
equation.
 We can also go the other way.  g f   x 
means g(f(x)).

Given
f(g(3)) =
= f(6)
= 4(6)2 – 2(6)
= 144 – 12
= 132
f ( x)  4 x  2 x
2
g(3) = 6
g ( x)  2 x
Given

f
f ( x) 
1)
 ff ((xx 1)

g (x)  x  1
x
g
g x 
 ff ((gg((xx))))
1
g(x) = x+1
1
f
x 
 g ( f ( x ))
1
 g 
x

x 1
1
1
x
Substitute x+1
In place of the
x in the f equation
The new x in the g
equation
=
The Chain Rule
Chain Rule Example
Use the chain rule to find Dx(x2 + 5x)8
 Let u = x2 + 5x
Another way to think
of it: The derivative of
 Let y = u8

dy
 dy   du 



dx  du   dx 
 8u

7
the outside times the
derivative of the inside
2 x  5 
2
 8 x  5x

7
 2x  5 
Chain Rule – You Try
Use the chain rule to find Dx(3x - 2x2)3
 Let u = 3x - 2x2
 Let y = u3

The derivative of the
outside times the
derivative of the inside
dy
 dy   du 



dx  du   dx 
 3u

2
3  4 x 
 3 3x  2x
2

2
3  4 x 
Chain Rule



Find the derivative of y = 4x(3x + 5)5
This is the Product Rule inside the Chain Rule.
Let u = 3x + 5; y = u5
4
5
4 x  5u (3)   (3x  5 ) ( 4 )


4
5
4 x  5(3x  5 ) (3)   4(3x  5 )



4 x 15(3x  5 )
4
4

 4(3 x  5 )
6 0 x(3x  5 )  4(3x  5 )
5
5
Chain Rule
 6 0 x (3x  5 )
4
 4 (3x  5 )
5
F a c to r o ut th e c o m m o n f a c to r
4
 4 (3x  5 ) 1 5 x  (3x  5 ) 
4
 4 (3x  5 ) (18 x  5 )
Chain Rule



 3x  2 
7
Find the derivative of
x 1
This is the Quotient Rule in the Chain Rule
Let u = 3x + 2; let y = u7



6
7
( x  1)  7 u (3)   (3x  2) (1)


( x  1)
2
6
7
( x  1)  7 ( 3x  2) (3)   (3x  2)


( x  1)
2
6
7
2 1  ( x  1)(3x  2)   (3x  2 )


( x  1)
2
Chain Rule

7
6
2 1  ( x  1)(3x  2)   (3x  2)


( x  1)
2
F a c to r o ut th e c o m m o n f a c to r
6



(3x  2) 2 1( x  1)  (3x  2) 
( x  1)
(3 x  2 )
6
 2 1x  2 1  3x  2 
( x  1)
(3x  2)
6
2
2
18 x  2 3 
( x  1)
2
4.4
Derivatives of Exponential Functions
Derivative of ex
Derivative of ax
x
Dx 3  (l n 3)3
x
Other Derivatives
Examples – Find the Derivative

y = e5x
 e
g( x )
 e
5x
(g '(x )
(5)  5 e
5x
Examples – Find the Derivative

y = 32x+1

g( x )
 g '( x )

2x 1
 (2)

 ln a a
 ln 3 3

 2 ln 3 3
2x 1
Example


dy
2
x 1
Find
if y  e
5x  2
dx
Use the product rule
y e
x
2
1



1
 x2  1  

 Dx  5 x  2  2   5 x  2  Dx  e





1
1
2
  5 x  2  2 (5 )
x 1
e
(2x )
2
5

2 5x  2
Example
y  e
 e

2
x 1
1


 Dx  5 x  2  2  


5
2
x 1
2 5x  2
5e
2
x 1
2 5x  2



5 x  2  Dx

2
x 1 

5 x  2  2x e



2
x 1 

5 x  2  2x e



 x2  1  
e



Example Continued
5e
2
x 1
2

x 1 

   2xe


2 5x  2  



5e
2
x 1
e
2
x 1
( 4 x )(5 x  2)
2 5x  2
e
2
x 1
 5  4 x (5 x  2) 
2 5x  2
e
2
x 1

  2 5x  2 
5x  2  

  2 5x  2 
2
20 x  8x  5
2 5x  2

Get a common
denominator to
add the 2 parts
together
4.5
Derivatives of Logarithmic Functions
Definition
Bases – a side note

Everything we do is in Base 10.
We count up to 9, then start over. We change our numbering
every 10 units.
1
11
21
Two
tens
2
12
22
and
3
13
23…
One
Ones
…one
group
of
Place
4
14
s
ten and
5
15
1, 2,
6
16
3…ones
7
17
8
18
9
19
10
20

Bases

The Yuki of Northern California used Base 8.
They counted up to 7, then started over. The numbering
changed every 8 units.
1
13
25
Two
eights
2
14
26
and
One
3
15
27…
Ones
…one
eight
Place
4
16
s
and
5
17
3…ones
6
20
7
21
So, 17 in Base 8 = 15 in Base 10
10
22
11
23
258 = 2 eights + 5 ones = 21
12
24

Bases


The Mayans used Base 20.
The Sumerians and people of Mesopotamia
used Base 60.
Definition
Example
Find f’(x) if f(x) = ln 6x
 Remember the properties of logs
 ln 6x = ln 6 + ln x

d
dx
(ln 6) 
 0
1
x

d
dx
1
x
(ln x )
Definitions
Examples – Find the Derivatives
y = ln 5x
 If g(x) = 5x, then g’(x) = 5

dy
dx

g '(x )
g(x )

5
5x

1
x
F’(x)
f(x) = 3x ln x2
 Product Rule

 d
2
2
f '( x )  (3x ) 
ln x   ln x (3)
 dx


 2x 
2
 3x  2   ln x (3)
x 

 6  3 ln x
2

