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