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
35 x
Ex. Find the derivative of f ( x) e
f ( x) e
3 5 x
.
d
3 5x
dx
5e35 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 72 x 2
12
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 qt 10,0001 0.5e
0.4 t 1
Find q24 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
14