Post on 16-Jan-2016
Chapter 4Techniques 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
1 12) or n n n ndx nx x nx
dx
1) 0 a constant or 0d
c c cdx
3) ( ) ( ) or ( ) ( )d d
cf x c f x cf x cf xdx dx
4) ( ) ( ) d
f x g x f x g xdx
Available Rules for Derivatives
2
( ) ( ) ( ) ( )6)
( ) ( )
f xd f x g x f x g x
dx g x g x
5) ( ) ( ) ( ) ( ) d
f x g x f x g x f x g xdx
Two More Rules
The product rule
The quotient rule
If f (x) and g (x) are differentiable functions, then we have
3 7 2If ( ) 2 5 3 8 1 , find ( )f x x x x x f x
2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x
9 7 6 4 230 48 105 40 45 80 2x x x x x x
The Product Rule - Example
Derivative of first
Derivative of Second
2
22
3 2 2 3 5( )
2
x x xf x
x
2
22
3 10 6
2
x x
x
Derivative of numerator
Derivative of denominator
The Quotient Rule - Example
2
3 5If ( ) , find ( )
2
xf x f x
x
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.
Example:
2 4 3 6x x
To compute a value, first you would evaluate the parentheses then multiply the results, so this can be treated as a product.
2 4 3 6 5x x x
To compute a value, the last operation would be to subtract, so this can be treated as a difference.
Calculation Thought ExperimentExample:
The Chain Rule
( ) ( )d du
f u f udx dx
The derivative of a f (quantity) is the derivative of f evaluated at the quantity, times the derivative of the quantity.
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
Generalized Power Rule
17) n nd duu n u
dx dx
Example: 1 22 23 4 3 4d d
x x x xdx dx
1 221
3 4 6 42
x x x
2
3 2
3 4
x
x x
72 1
If ( ) find ( )3 5
xG x G x
x
6
2
3 5 2 2 1 32 1( ) 7
3 5 3 5
x xxG x
x x
66
2 8
91 2 12 1 137
3 5 3 5 3 5
xx
x x x
Generalized Power Rule
Example:
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
5 2 8 2If and 7 3 , find dy
y u u x x ydx
dy dy du
dx du dx 3 2 75
56 62
u x x
3 28 2 757 3 56 6
2x x x x
3 27 8 2140 15 7 3x x x x
Sub in for u
Logarithmic Functions
1ln 0
dx x
dx x
1ln
d duu
dx u dx
Generalized Rule for Natural Logarithm Functions
Derivative of the Natural Logarithm
If u is a differentiable function, then
Find the derivative of 2( ) ln 2 1 .f x x
1( ) 2f x
x
Find an equation of the tangent line to the graph of ( ) 2 ln at 1, 2 .f x x x
2
2
2 1( )
2 1
dx
dxf xx
2
4
2 1
x
x
(1) 3f
2 3( 1)
3 1
y x
y x
Slope: Equation:
Examples
1log
lnbd
xdx x b
1log
lnbd du
udx u b dx
Generalized Rule for Logarithm Functions.
Derivative of a Logarithmic Function.
If u is a differentiable function, then
More Logarithmic Functions
4log 2 3 4d
x xdx
4 4log 2 log 3 4d
x xdx
1 1( 4)
( 2) ln 4 (3 4 ) ln 4x x
Examples
Logarithms of Absolute Values
1log
lnbd du
udx u b dx
1ln
d duu
dx u dx
2ln 8 3d
xdx
2
116
8 3x
x
31
log 2d
dx x
2
1 1
1/ 2 ln 3x x
2
1
2 ln 3x x
Examples
Exponential Functions
x xde e
dx
u ud due e
dx dx
Generalized Rule for the natural exponential function.
Derivative of the natural exponential function.
If u is a differentiable function, then
Find the derivative of 3 5( ) .xf x e
4 4( ) xf x x e
3 44 1xx e x
Find the derivative of 4 4( ) xf x x e
3 5( ) 3 5x df x e x
dx 3 55 xe
3 4 4 44 4x xx e x e
4 4 4 4x xx e x e
Examples
lnx xdb b b
dx
lnu ud dub b b
dx dx
Generalized Rule for general exponential functions.
Derivative of general exponential functions.
If u is a differentiable function, then
Exponential Functions
Find the derivative of 2 2( ) 7x xf x
2 22 2 7 ln 7x xx
2 2 2( ) 7 ln 7 2x x df x x x
dx
Exponential Functions