Transcript Chapter Four Review Some Practice Problems

Section 5.1
The Natural Log Function:
Differentiation
Section 5.1 The Natural Log Function:
Differentiation


Since the winter break we have spent our
mental energy on integrating functions. We
have looked at the integral as an infinite
sum, as an antiderivative of a function, and
as an area bounded by a curve, the x-axis,
and vertical boundaries.
Now, we will step back and look at
differentiation rules for logarithmic and
exponential functions before we deal with
antidifferentiating them.
Section 5.1 The Natural Log Function:
Differentiation

You are familiar with the power rule for
integrals which states that
n 1
x
n
x
 dx  n  1  C

This definition will clearly cause us problems
if we try to apply it to the simple function of
1
y
x
Section 5.1 The Natural Log Function:
Differentiation
We need a new definition to deal with this
function.
x
1
ln x   dt
t
1
Let’s take out our calculator and see if we
can verify this definition. Any ideas on how to
do so?
Section 5.1 The Natural Log Function:
Differentiation
 We should remind ourselves of the
important properties of the natural logarithm
function as well as remember where we have
used the function in the past. The Laws of
Logarithms:
ln a  ln b  ln(ab)
a
ln a  ln b  ln  
b
ln a b  b ln a
Section 5.1 The Natural Log Function:
Differentiation
 We should also be reminded of a special
number that comes into play when dealing
with natural logarithms. That number is
called e. You have seen a definition of e in the
past. That definition is
n
 1
1    e
lim
n
n 
 We have a new definition now
e
1
1 x dx  1
Section 5.1 The Natural Log Function:
Differentiation

We can use the properties of logarithms to
help clean up some ugly looking
differentiation problems in addition to
practicing new integrals based on our
definitions. The Fundamental Theorem of
Calculus tells us that
x
1
d
1
1 t dt  ln x  dx  ln x   x
Section 5.1 The Natural Log Function:
Differentiation

Adding the Chain Rule to that application of
the Fundamental Theorem of Calculus we
have the following derivative statement and
an example of its application:
d
1 du
 ln u  
dx
u dx
dy
2x  3
y  ln  x  3x  1 
 2
dx x  3x  1
2
Section 5.1 The Natural Log Function:
Differentiation

Here are a couple of more examples to try. See how
you might use the properties of logarithms to help
simplify the differentiation process.
dy
y  x ln x 

dx
3
y
y
3
x 1
dy


x 1
dx
x 2 3x  2
 x  1
2
dy


dx