Lesson 5 Derivatives of Logarithmic Functions and ...

Course II

Lesson 5 Derivatives of Logarithmic Functions

and Exponential Functions

5A

? Derivative of logarithmic functions

1

Review of the Logarithmic Function

Exponential function

y

y = ax

y = ax (a > 0, a 1)

Fig.1

Fig.1

O

x

Logarithmic function

y = ax

y

y = ax

Fig.2

Fig.1

x

We replace the notation

x = ay

y = loga x

Fig.3

y

y = loga x

O

x

2

Derivative of the Logarithmic Function

From the definition

(log a

x)

=

lim

h0

loga

(x

+

h)

h

-

loga

x

=

lim

h0

1 h

loga 1 +

h x

=

lim

h0

1 x

x h

loga 1 +

h x

We put

h =k x

.

When h 0, k 0.

(log a

x)

=

lkim0

1 x

1 k

loga

(1

+

k

)

?

=

1 x

lim

k 0

log

a

(1

+

k

1

)k

=

1 x

log

a

lkim0(1

+

k

1

)k

3

Napier's Constant

Trial

We expect that

approaches one value as

.

Napier's Constant

e

=

lim1 + n

1 n

n

=

Important Mathematical Constants

"=3.1415..." was known 4000 years ago

"e=2.7182..." was found in 17th century

4

Natural Logarithm

Then

If the base is e, we have

Natural logarithm is the logarithm to the base e.

Notation: loge x ln x

Summary

d

1

d

1

dx (loga x) = x ln a ,

(ln x) =

dx

x

5

1

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download