Transcript 3.6

3.6 – Derivatives of Logarithmic
Functions
1
Rules
Basic Rule :
d  log a x 
dx
1

x ln a
With Chain Rule :
d log a u 
dx
1 du


u ln a dx
d ln x  1
Basic Rule :

dx
x
Why is the
absolute value
needed?
2
Rules
Basic Rule :
d  a x 
dx
Chain Rule :
 a x  ln a
d  au 
dx
du
 a  ln a 
dx
u
3
Try Theses: Basics
Evaluate the following.
d 3x
e  
1.
dx
d x
7  
2.
dx
d
3.  log 3 x 
dx
d
4.
ln  2 x   
dx
4
Try Theses
Determine the first derivative of each of the
following. Do only minor simplification.
1. g ( x)  log 3  cos x 
2. f ( x)  ln  x sin x 
4
3. y  sec  3   ln  tan x 
x
4. h  x  
e
2
csc x
x 2 3 x
5
Laws of Logarithms
1. log a  xy   log a x  log a y
x
2. log a    log a x  log a y
 y
3. log a  x y   y  log a x
4. a loga x  x or log a a x  x
6
Logarithmic Differentiation
1. Take the natural logarithms of both sides of an
equation y = f(x).
2. Use the laws of logarithms to separate expressions.
Be sure to create a single product or quotient on the
right-hand side at this stage.
3. Differentiate implicitly with respect to x.
4. Solve the resulting equation for y′.
7
Examples
Determine the first derivative of each of the
following.
1. y 
x
e
5
x3  7 x
cos x
 9 x  3
2
97
2. f  x   x1/ x
8
Try Theses
Determine the first derivative of each of the
following.
1. y  xe x  x 2  1
10
2
2. g  x  
4
x2  1
x2 1
3. h  x    sin x 
4. y   ln x 
cos x
csc x
9