Chapter 4
Techniques of
Differentiation
Sections 4.1, 4.2, and 4.3
Techniques of Differentiation
 The Product and Quotient Rules
 The Chain Rule
 Derivatives of Logarithmic and Exponential
asFunctions
Available Rules for Derivatives
1)
d
c  0
dx
2)
d n
x  nx n1
dx
3)
d
d
 cf ( x)   c  f ( x) 
dx
dx
4)
d
 f  x   g  x    f ( x)  g ( x)
dx
 
 c a constant 
or
or
 
x
n

 c   0
 nx n 1
or
 cf ( x)   cf ( x)
Two More Rules
If f (x) and g (x) are differentiable functions, then
we have
The product rule
5)
d
 f  x   g  x    f ( x ) g ( x )  f ( x ) g ( x )
dx
The quotient rule
6)
d  f  x   f ( x ) g ( x )  f ( x ) g ( x )



2
dx  g ( x ) 
 g ( x )
The Product Rule - Example



If f ( x )  x3  2 x  5 3x7  8 x 2  1 , find f ( x )


 

2
7
2
3
6

f ( x)  3x  2 3x  8 x  1  x  2 x  5 21x  16 x
Derivative
of first
Derivative
of Second
 30 x  48x  105x  40 x  45x  80x  2
9
7
6
4
2

The Quotient Rule - Example
3x  5
If f ( x )  2
, find f ( x )
x 2
Derivative of
numerator
f ( x) 



3 x 2  2  2 x  3x  5
x
2
2

3x2  10 x  6
x
2
2

2
2
Derivative of
denominator
Calculation Thought Experiment
Given an expression, consider the steps you
would use in computing its value. If the last
operation is multiplication, treat the expression
as a product; if the last operation is division,
treat the expression as a quotient; and so on.
Calculation Thought Experiment
Example:
 2x  43x  6
To compute a value, first you would evaluate the
parentheses then multiply the results, so this can
be treated as a product.
Example:  2x  43x  6  5x
To compute a value, the last operation would be
to subtract, so this can be treated as a difference.
The Chain Rule
If f is a differentiable function of u and u is a
differentiable function of x, then the composite
f (u) is a differentiable function of x, and
d
du
 f (u )  f (u )
dx
dx
The derivative of a f (quantity) is the derivative of f
evaluated at the quantity, times the derivative of the
quantity.
Generalized Power Rule
7)
Example:
d n
n 1 du
u  nu
dx
dx


12
d 
d
2
2

3x  4 x 
3x  4 x
 dx
dx 
1 2
1
2
 3x  4 x
6x  4
2


3x  2
3x 2  4 x

Generalized Power Rule
7
Example: If G( x)   2 x  1  find G( x)
 3x  5 
6
 2 x  1    3x  5 2   2 x  1 3 
G( x)  7 
 
2

3
x

5

 
 3x  5 

 2x 1 
 7

 3x  5 
6
13
 3x  5
2

91 2 x  1
 3x  5
8
6
Chain Rule in Differential
Notation
If y is a differentiable function of u and u is a
differentiable function of x, then
dy dy du


dx du dx
Chain Rule Example
dy
If y  u and u  7 x  3 x , find y  
dx
dy dy du 5 3 2


 u  56 x 7  6 x
dx du dx 2
52
8
2


Sub in for u

 

 15x  7 x  3x 
5
 7 x8  3 x 2
2

 140 x
7
32
 56 x 7  6 x
8
2
32
Logarithmic Functions
Derivative of the Natural Logarithm
d
1
ln x 
dx
x
 x  0
Generalized Rule for Natural Logarithm Functions
If u is a differentiable function, then
d
1 du
ln u 
dx
u dx
Examples


Find the derivative of f ( x)  ln 2 x  1 .
d  2 
2 x  1
4x

dx
 2
f ( x) 
2
2x 1
2x 1
Find an equation of the tangent line to the graph of
2
f ( x)  2x  ln x at 1,2.
Slope:
1
f ( x)  2 
x
f (1)  3
Equation:
y  2  3( x  1)
y  3x  1
More Logarithmic Functions
Derivative of a Logarithmic Function.
d
1
log b x 
dx
x ln b
Generalized Rule for Logarithm Functions.
If u is a differentiable function, then
d
1 du
logb u 
dx
u ln b dx
Examples
d
log 4   x  2  3  4 x  
dx
d
log 4  x  2   log 4  3  4 x  

dx
1
1


(4)
( x  2) ln 4 (3  4 x) ln 4
Logarithms of Absolute Values
d
1 du
ln u 
dx
u dx
d
1 du
logb u 
dx
u ln b dx
Examples
d
1
2
ln 8 x  3  2
16 x 
dx
8x  3
d
1
1
 1 
log3  2 
 2

dx
x
1/ x  2 ln 3  x 
1

x  2 x 2 ln 3


Exponential Functions
Derivative of the natural exponential function.
d x
e   e x
dx  
Generalized Rule for the natural exponential function.
If u is a differentiable function, then
d u
du
u
e   e
dx  
dx
Examples
35 x
f
(
x
)

e
.
Find the derivative of
f ( x)  e
3 5 x
d
35 x
 3  5 x   5e
dx
Find the derivative of f ( x)  x e
4 4x

f ( x)  x e
4 4x
  
 

4  4x
4
4x 
 x e x e
 4 x e  4 x e  4 x3e4 x 1  x 
3 4x
4 4x
Exponential Functions
Derivative of general exponential functions.
d x
b   b x ln b
dx  
Generalized Rule for general exponential functions.
If u is a differentiable function, then
d u
du
u
b   b ln b
dx  
dx
Exponential Functions
Find the derivative of f ( x)  7
f ( x)  7
x2  2 x
x2  2 x

d 2
ln 7
x  2x
dx
  2x  2 7
x2  2 x
ln 7
