Download Techniques of Differentiation

Techniques of Differentiation
• The Product and Quotient Rules
• The Chain Rule
• Derivatives of Logarithmic and Exponential
Functions
• Implicit Differentiation
The Product Rule
d
 f  x   g  x    f ( x) g ( x)  f ( x) g ( x)
dx
The Quotient Rule
d  f  x   f ( x) g ( x)  f ( x) g ( x)


2
dx  g ( x) 
 g ( x) 
The Product Rule
Ex. f ( x)   x3  2 x  5 3x7  8x 2  1


 

f ( x)  3x 2  2 3x7  8 x 2  1  x3  2 x  5 21x6  16 x
Derivative
of first
Derivative of
Second
 30x9  48x7  105x6  40x4  45x2  80x  2

The Quotient Rule
3x  5
Ex. f ( x)  2
x 2
Derivative of
numerator
f ( x) 


Derivative of
denominator

3 x 2  2  2 x  3x  5
x
2
2

2
3x 2  10 x  6
x
2
2

2
Compute the Derivative

d 
3
Ex.
x  3x
dx 

2

 1 9x

 x
2
 x
2

 2x 
 x 1
 
 2 x  x  3x
3
  2x  2
 1  9 1  2   1  3 2  2  
= –10
x 1
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
Ex.  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.
Ex.  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
d n
n 1 du
u  n u
dx
dx


12
d 
d
2
2

3x  4 x 
3x  4 x
Ex.
 dx
dx 
1 2
1
2
 3x  4 x
6x  4
2
3x  2

3x 2  4 x


The Chain Rule
2x 1 

Ex. G ( x)  

 3x  5 
7
2 x  1   3x  5 2   2 x  1 3 


G( x)  7 
 
2

 3x  5  
 3x  5 

 2x 1 
 7

 3x  5 
6
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
Ex. y  u
52
, u  7 x  3x
8
2
dy dy du


dx du dx


5 32
 u  56 x 7  6 x
Sub in for u
2
32
5
8
2
 7 x  3x
 56 x 7  6 x
2


 140 x
7
 

 15 x  7 x  3x 
8
2
32
Differentiation of 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


Ex. Find the derivative of f ( x)  ln 2 x  1 .
d  2 
2 x  1

4x
dx
f ( x) 
 2
2
2x 1
2x 1
2
Ex. Find an equation of the tangent line to the graph of
f ( x)  2 x  ln x at 1, 2 .
1
f ( x)  2 
x
f (1)  3
Slope:
Equation:
y  2  3( x  1)
y  3x  1
Differentiation of Logarithmic
Functions
Derivative of a Logarithmic Function:
d
1
logb x 
dx
x ln b
Generalized Rule for Logarithm Functions
If u is a differentiable function, then
d
1 du
log b u 
dx
u ln b dx
Differentiation of Logarithmic
Functions
d
log 4   x  2  3  4 x  
Ex.
dx
d
log 4  x  2   log 4  3  4 x  

dx
1
1


(4)
( x  2) ln 4 (3  4 x) ln 4
Derivative of Logarithms of
Absolute Values
d
1 du
ln u 
dx
u dx
d
1 du
logb u 
dx
u ln b dx
Derivative of Logarithms of
Absolute Values
d
1
2
Ex.
ln 8 x  3  2
16 x 
dx
8x  3
d
1
1
 1 
log3  2 
 2
Ex.

dx
x
1/ x  2  ln 3  x 
1

2
x  2 x ln 3


Differentiation of Exponential
Functions
Derivative of ex:
d x
e   e x
dx  
Generalized Rule for eu:
If u is a differentiable function, then
d u
du
 e   eu
dx  
dx
Derivatives of Exponential
Functions
35 x
f
(
x
)

e
.
Ex. Find the derivative of
f ( x)  e
35 x
d
3  5x 
dx
35 x
 5e
4 4x
f
(
x
)

x
e
Ex. Find the derivative of
f ( x)  4 x3e4 x  4 x 4e4 x
 4 x3e4 x 1  x 
Differentiation of Exponential
Functions
Derivative of bx:
d x
b   b x ln b
dx  
Generalized Rule for bu:
If u is a differentiable function, then
d u
du
b   bu ln b
dx  
dx
Derivatives of Exponential
Functions
Ex. Find the derivative of f ( x)  7
f ( x)  7
7
x2  2 x
x2  2 x
x2  2 x

.
d 2
x  2x
dx
 2 x  2

Implicit Differentiation
y  3x  4 x  17
3
y is explicitly a function of x.
y  xy  3x  1
3
y is implicitly a function of x.
Implicit Differentiation (cont.)
To differentiate the implicit case we use the chain rule
where y is a function of x:
dy
dy
3y
 yx
3
dx
dx
dy
2
3y  x  3  y
dx
dy
3 y
 2
dx 3 y  x
2


dy
Solve for
dx
Tangent Line to Implicit Curve
Ex. Find the equation of the tangent line to
x
3
the curve y  ln y 
at the point (2, 1).
2
dy 1 dy 1
3y


dx y dx 2
2
dy
dy 1
3 1 
dx
dx 2
dy 1

dx 8
1
y 1   x  2
8
Logarithmic Differentiation
Ex. Use logarithmic differentiation to find the
derivative of y  3x  2  9 x  15 .
ln y  ln

3 x  2  9 x  1
5

Apply ln
1
ln y  ln  3x  2   5ln  9 x  1 Properties of ln
2
1 dy
3
5(9)


y dx 2  3x  2  9 x  1
Differentiate
dy
3
45 
5
 3x  2  9 x  1 



dx
 2  3x  2  9 x  1 
Solve