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Lesson: Derivative Applications 3
 Objective – Logarithms,
Euler’s Number,
& Differentiation,
oh my!
1. What is a Logarithm?
A logarithm can be defined as an exponent
Consider the following logarithmic expression
Log7 49
It represents the exponent that the base of 7
must be raised to, in order to get the value 49.
Log7 49  2
2. What is Euler’s Number?
e  2.718182...
e is a real number constant that appears in some kinds of
mathematics problems. Examples of such problems are those
involving growth or decay (including compound interest), the
statistical "bell curve," the shape of a hanging cable (or the
Gateway Arch in St. Louis), some problems of probability,
some counting problems, and even the study of the distribution
of prime numbers. It appears in Stirling's Formula for
approximating factorials. It also shows up in calculus quite
often, wherever you are dealing with either logarithmic or
exponential functions. There is also a connection between e
and complex numbers, via Euler's Equation.
The number e was first studied by the Swiss
mathematician Leonhard Euler in the 1720s,
although its existence was more or less implied in
the work of John Napier, the inventor of logarithms,
in 1614. Euler was also the first to use the letter e for
it in 1727 (the fact that it is the first letter of his
surname is coincidental). As a result, sometimes e is
called the Euler Number, the Eulerian Number, or
Napier's Constant.
3. What is the Natural Logarithm (LN)?
A logarithm with a base of e
4. Properties of Logs
Logb x  Logb y  Logb ( xy)
x
Logb x  Logb y  Logb  
 y
n
Logb x  n  Logb x
Formulas:
d
1
ln( x) 
dx
x
d x
x
e   e
dx
d
1
logb ( x) 
dx
x  ln b
 Basically, think of derivatives of “e” and
“Ln” as chain rule problems.


d
1
( stuff ) '
 ( stuff ) ' 
 Ln(stuff ) 
dx
stuff
stuff
d ( stuff )
e
  e( stuff )  ( stuff ) '
dx
 Let’s try a few:
d 5x
5x
5x
e   e  5  5e
1.
dx
d  x2 
2.
e
dx  
 e  2x  2x  e
x2
d  5 x 3 
3

5
x
2
3.
e

e


15
x


dx
 15 x  e
2
5 x3
x2
1
1
d
7
4.

 Ln(7 x)   7 
dx
7x
x
7x
d
1
3
3
2


5.
Ln( x )  

3x


3
dx
x
x
d
1
6.
  sin( x)
 Ln(cos( x)) 
dx
cos( x)
 sin( x)
  tan( x)

cos( x)
What’s up with ex? How can it be the derivative of itself?
d x
x
e   e
 Proof:
dx
 Let’s start with:
 LN both sides:
 Use log property:
 Simplify:
ye
x
Ln( y)  Ln(e )
x
Ln( y )  x  Ln(e)
Ln( y )  x 1
 Derive both sides
with respect to x:
Ln( y )  x
1
 y' 1
y
 Multiply both sides
by y:
1
y   y '  1 y
y
 Simplify:
 Simplify:
y' y
 Replace y with the
original equation:
y'  e
x
Q.E.D.
d
2


ln(
x

1)
EX. 1: Find dx 

1
d 2
 2
  x  1
x  1 dx
1
 2
 2x
x 1
2x
 2
x 1
d
3
EX. 2: Find
ln(4 x  5 x  3) 
dx
2

 x  Sinx  
EX. 3: Find d
ln 

dx   1  x  
If the mere sight of this problem makes you
want to break stuff & cry don’t worry. You
are not alone.
But fear not, it is very workable with the use
of the logarithmic properties.
Now isn’t that special?
2

 x  Sinx  
d
ln 

dx   1  x  
d 
2


ln(
x
)

ln(
Sinx
)

ln(
1

x
)

dx 

d 
 2  ln( x)  ln( Sinx)  ln(1  x) 
dx 

1
2
d 
1


2

ln(
x
)

ln(
Sinx
)


ln(1

x
)


dx 
2

1
1
1 1
 2 
 Cosx  
x Sinx
2 1 x
2 Cosx
1
 

x Sinx 2(1  x)
2
1
  Cotx 
x
2  2x
x  7 x  14
y
2 4
(1  x )
 x 2  3 7 x  14 
ln y  ln 

2 4
(1

x
)


2
EX. 4: Find y’, if
1
3
3
ln y  ln( x )  ln(7 x  14)  ln(1  x )
2
2 4
1
2
ln( y )  2  ln( x)   ln(7 x  14)  4  ln(1  x )
3
1
2
ln( y )  2  ln( x)   ln(7 x  14)  4  ln(1  x )
3
1
1 1
1
1
 y '  2  
7  4

2
x
2
y
x 3 7 x  14
1 x
1
2
7
8x
 y'  

2
y
x 3(7 x  14) 1  x
1
 2
7
8x 
y   y '   


y

2
y
  x 3(7 x  14) 1  x 
2
7
8x 
y '  y  

2 
 x 3(7 x  14) 1  x 
x  7 x  14  2
7
8x 
y' 
 

2 4
2 
(1  x )
 x 3(7 x  14) 1  x 
2
3
d
EX. 5: Find
ln(Cotx)
dx
d
ln(ln
x
)


EX. 6: Find dx
Deriving the exponential function
 Remember:
e
x
d x
x
e   e
dx
Stuff
 What about
d 3x
3x
 e   3e
dx
Ex. 7: Differentiate
d 5x
e  
A.
dx
d 12 x
e  
B.
dx
d 4 x
e  
C.
dx
d  3 x2 
D.
e


dx 
d 
7 x5 
E.
4e


dx 
d  x
F.
e

dx  
HW 4.2: Log Differentiation Worksheet