Transcript Document

Lecture 11
§4.3 Derivatives of Logarithmic and
Exponential Functions (Pages 261~272)
Derivatives of Logarithms of Functions:
Original Rule
d
1
ln x 
dx
x
d
1
logb x 
dx
x ln b
Generalized Rule
d
1 du
ln u 
dx
u dx
d
1 du
logb u 
dx
u ln b dx
1
Derivatives of Logarithms of Absolute Values:
Original Rule
Generalized Rule
d
1

ln x  
dx
x
d
1
logb x 
dx
x ln b
d
1 du

ln u  
dx
u dx
d
1 du
logb u 
dx
u ln b dx




Derivatives of Exponentials of Functions:
Original Rule
 
d x
e  ex
dx
d x
b  b x ln b
dx
 
Generalized Rule
 
d u
u du
e e
dx
dx
d u
du
u
b  b ln b
dx
dx
 
2
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
3
Examples


Ex. Find the derivative of f ( x)  ln 2 x  1 .
d  2 
2 x  1
4x

dx

f ( x) 
2
2
2
x
1
2x 1
2
Ex. Find an equation of the tangent line to the graph of
f ( x)  2x  ln x at 1,2.
1
f ( x)  2 
x
f (1)  3
Slope:
y  2  3( x  1)
Equation:
y  3x  1
5
y = 2x + lnx
4
y = 3x -1
3
2
y 1
0
-1
0
0.5
1
1.5
2
-2
-3
x
4
Differentiation of 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
5
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
6
Derivative of Logarithms of Absolute Values
Original Rule
d
1

ln x  
dx
x
d
1
logb x 
dx
x ln b


Generalized Rule
d
1 du

ln u  
dx
u dx
d
1 du
logb u 
dx
u ln b dx


These two
graphs show
that
 1 1
d
ln x    
dx
 x x
7
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


8
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
9
Derivatives of Exponential Functions
35 x
Ex. Find the derivative of f ( x)  e
f ( x)  e
3 5 x
.
d
3  5x 
dx
 5e35 x
Ex. Find the derivative of f ( x)  x e
4 4x
3 4x
4 4x

f ( x)  4x e  4x e
 4 x3e4 x 1  x 
10
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
u
b   b ln b
dx  
dx
11
Derivatives of Exponential Functions
Ex. Find the derivative of f ( x)  7
f  x   7
x 2 2 x
7
x2  2 x
.

d 2
ln 7 x  2 x
dx
x 2 2 x

ln 72 x  2
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Example 5:
Sales Growth
The sales of the Cyberpunk II
arcade video game can be modeled by the logistic
curve
10,000
q t  
1  0.5e  0.4 t
Where q(t) is the total number of units sold t months
after its introduction. How fast is the game selling 2
years after its introduction?
Solution:

Given qt   10,0001  0.5e


0.4 t 1
Find q24  10,0001  0.5e
 0.5e  0.4
0.4 t 2
0.4 t
t 24
 0.135units per month
13
HW Problem 83:
Radioactive Decay
Plutonium-239 has a half-life
of 24,400 years. How fast is a lump of 10 grams
decaying after 100 years?
Hints:
Radioactive decay : N t   N oe kt  10e 0.0000284t
dN
kt d
Then
 N oe .  kt   0.000284e 0.0000284t
dt
dt
dN
T hus
dt
g
(interpret your numercial answer)
t 100  ?
year
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