Upload
kelley-greene
View
230
Download
0
Tags:
Embed Size (px)
Citation preview
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