Download - Lesson: Derivative Applications 3 Objective – Logarithms, Euler’s Number, & Differentiation, oh my!

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Lesson: Derivative Applications 3

Objective – Logarithms, Euler’s Number, & Differentiation, oh my!

1. What is a Logarithm?

7 49Log

A logarithm can be defined as an exponent

Consider the following logarithmic expression

It represents the exponent that the base of 7must be raised to, in order to get the value 49.

7 49 2Log

2. What is Euler’s Number?

2.718182...ee 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)?

( )b b bLog x Log y Log xy

A logarithm with a base of e

4. Properties of Logs

b b b

xLog x Log y Log

y

n

b bLog x n Log x

Formulas:

1ln( )

dx

dx x

x xde e

dx

1log ( )

lnb

dx

dx x b

Basically, think of derivatives of “e” and “Ln” as chain rule problems.

1( ) ( ) '

dLn stuff stuff

dx stuff

( ) 'stuff

stuff

( ) ( ) ( ) 'stuff stuffde e stuff

dx

Let’s try a few:

51. xde

dx

5 5xe 55 xe

2

2. xde

dx

2

2xe x 2

2 xx e

353. xde

dx

35 215xe x

32 515 xx e

4. (7 )dLn x

dx

17

7x 1

x

35. ( )dLn x

dx

23

13x

x 3

x

6. (cos( ))dLn x

dx

7

7x

1sin( )

cos( )x

x

sin( )

cos( )

x

x

tan( )x

What’s up with ex? How can it be the derivative of itself?

Proof:x xde e

dx

Let’s start with: xy e LN both sides: ( ) ( )xLn y Ln e

Use log property: ( ) ( )Ln y x Ln e

Simplify: ( ) 1Ln y x

Derive both sides with respect to x:

Multiply both sides by y:

Simplify:

Simplify: ( )Ln y x1

' 1yy

1' 1y y y

y

'y y

Replace y with the original equation:

' xy eQ.E.D.

EX. 1: Find2ln( 1)

dx

dx

22

11

1

dx

x dx

2

12

1x

x

2

2

1

x

x

EX. 2: Find 3ln(4 5 3)d

x xdx

EX. 3: Find2

ln1

d x Sinx

dx x

Now isn’t that special?

If the mere sight of this problem makes youwant 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.

2

ln1

d x Sinx

dx x

2ln( ) ln( ) ln( 1 )d

x Sinx xdx

1

22 ln( ) ln( ) ln(1 )d

x Sinx xdx

12 ln( ) ln( ) ln(1 )

2

dx Sinx x

dx

1 1 1 12

2 1Cosx

x Sinx x

2 1

2(1 )

Cosx

x Sinx x

2 1

2 2Cotx

x x

EX. 4: Find y’, if2 3

2 4

7 14

(1 )

x xy

x

2 3

2 4

7 14ln ln

(1 )

x xy

x

12 2 43ln ln( ) ln(7 14) ln(1 )y x x x

21ln( ) 2 ln( ) ln(7 14) 4 ln(1 )

3y x x x

21ln( ) 2 ln( ) ln(7 14) 4 ln(1 )

3y x x x

2

1 1 1 1 1' 2 7 4 2

3 7 14 1y x

y x x x

2

1 2 7 8'

3(7 14) 1

xy

y x x x

2

1 2 7 8'

3(7 14) 1

xy y y

y x x x

2

2 7 8'

3(7 14) 1

xy y

x x x

2 3

2 4 2

7 14 2 7 8'

(1 ) 3(7 14) 1

x x xy

x x x x

EX. 5: Find ln( )d

Cotxdx

EX. 6: Find ln(ln )d

xdx

Deriving the exponential function

Remember:

What about

xe

x xde e

dx

3xde

dx

Stuff

33 xe

5. xdA edx

Ex. 7: Differentiate

4. xdC edx

12. xdB edx

57. 4 xdE edx

. xdF edx

23. xdD edx

HW 4.2: Log Differentiation Worksheet