derivatives of log functions
DESCRIPTION
Derivatives of Log Functions. Lesson 4.5. Problem. Consider f(x) = log a x 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. Possible Solution. We know that the derivative is the "slope function" - PowerPoint PPT PresentationTRANSCRIPT
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
9