Download - Derive Exponential Derivative Rule
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