derive exponential derivative formula
TRANSCRIPT
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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????
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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
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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
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So the derivative of 2x is proportional to 2x with a constant of proportionality 0.693.
Hmmmm…
693.02)2( xx
dx
d
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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????
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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
8 2.079
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