The Derivatives of Exponential Functions

Calculate the derivative of f(x)=2x

0 0 0

0

2 2 2 2 2 2 (2 1)'( ) lim lim lim

2 1'( ) 2 lim

x h x x h x x h

h h h

hx

h

f xh h h

f xh

The Derivatives of Exponential Functions

Calculate the derivative of f(x)=2x

0 0 0

0

2 2 2 2 2 2 (2 1)'( ) lim lim lim

2 1'( ) 2 lim

x h x x h x x h

h h h

hx

h

f xh h h

f xh

What is this????

The Derivatives of Exponential Functions

Fill out the following table for values of h close to zero.

h-0.01

-0.001-0.00010.00010.0010.01

0

2 1lim

h

h h

The Derivatives of Exponential Functions

Fill out the following table for values of h close to zero.

h-0.01 .69075

-0.001 .69291-0.0001 .693120.0001 .693170.001 .693390.01 .69556

0

2 1lim

h

h h

The Derivatives of Exponential Functions

Fill out the following table for values of h close to zero.

h-0.01 .69075

-0.001 .69291-0.0001 .693120.0001 .693170.001 .693390.01 .69556

0

2 1lim

h

h h

This table suggests that the limit DOES

exist, and has a value of about 0.693

The Derivatives of Exponential Functions

Fill out the following table for values of h close to zero.

h-0.01 .69075

-0.001 .69291-0.0001 .693120.0001 .693170.001 .693390.01 .69556

0

2 1lim

h

h h

This table suggests that the limit DOES

exist, and has a value of about 0.693

So we can write: 693.02)2( xx

dx

d

So the derivative of 2x is proportional to 2x with a constant of proportionality 0.693.

Hmmmm…

693.02)2( xx

dx

d

The Derivatives of Exponential Functions

Calculate the derivative of f(x)=ax

h

aaxf

h

aa

h

aaa

h

aaxf

h

h

x

hx

h

xhx

h

xhx

h

1lim)('

)1(limlimlim)('

0

000

The Derivatives of Exponential Functions

Calculate the derivative of f(x)=ax

h

aaxf

h

aa

h

aaa

h

aaxf

h

h

x

hx

h

xhx

h

xhx

h

1lim)('

)1(limlimlim)('

0

000

What is this????

Here is for different values of a0

1lim

h

h

a

h

a

2 0.693

3 1.0986

4 1.386

5 1.609

6 1.797

7 1.946

0

1lim

h

h

a

h

( )xda

dx

693.02)2( xx

dx

d

Here is for different values of a0

1lim

h

h

a

h

a

2 0.693

3 1.0986

4 1.386

5 1.609

6 1.797

7 1.946

0

1lim

h

h

a

h

Use your calculator to plot these points.

What type of function does it look

like?

( )xda

dx

693.02)2( xx

dx

d

Here is for different values of a0

1lim

h

h

a

h

a

2 0.693

3 1.0986

4 1.386

5 1.609

6 1.797

7 1.946

0

1lim

h

h

a

h

Turns out that

the graph is just

y = ln(a)

( )xda

dx

693.02)2( xx

dx

d

Here is for different values of a0

1lim

h

h

a

h

a

2 ln(2) = 0.693

3 ln(3) = 1.0986

4 ln(4) = 1.386

5 ln(5) = 1.609

6 ln(6) = 1.797

7 ln(7) = 1.946

0

1lim

h

h

a

h

( )xda

dx

)2ln(2)2( xx

dx

d

)3ln(3)3( xx

dx

d

)4ln(4)4( xx

dx

d

)5ln(5)5( xx

dx

d

)6ln(6)6( xx

dx

d

)7ln(7)7( xx

dx

d