Derivatives of Log Functions

Lesson 4.5

Problem

Consider f(x) = logax

What if we try to use the definition for derivative using the limit

No way to break up this portion of the expression to let h → 0 2

0

log ( ) loglim a a

h

x h x

h

Possible Solution

We know that the derivative is the "slope function"

What if we graph y=ln(x) and check the slopes … plotting them

3

Slope Results

The table at the right shows the values of theslopes at various x values

What function might this be?

Appears to be

4

xslope of ln(x) at x

0.001 1000.000

0.010 100.000

0.100 10.000

0.500 2.000

0.750 1.333

1.000 1.000

1.500 0.667

2.000 0.500

5.000 0.200

10.000 0.100

1y

x

Derivative of the Log Function

For the natural logarithm ln(x)

•

For the log of a different base loga(x)

•

5

1lnxD x

x

1

loglnx aD xa x

Examples

Try these sample problems … find the derivative• Don't forget to use the chain rule where

applicable

6

2ln 1y x

( ) 3 1 ln 1f x x x

3/ 24 2( ) ln 5f x x x

What About ln(-x)?

Consider it a compound function

Apply the chain rule

Thus we see 7

( ) ln( ) ( )

( ( ))

f x x g x x

y f g x

1 ( ) 1 1ln( ) 1x

d xD x

x dx x x

ln( ) ln( )x xD x D x

Conclusion

We now can say

Apply to finding these derivatives

8

1 1

ln loglnx x aD x D x

x a x

ln 4y x

( ) ln 3f x x

5log 5 2y x

Assignment

Lesson 4.5

Page 289

Exercises 1 – 65 EOO

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