derivatives of exponential and logarithmic functions
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Derivatives of exponential and logarithmic functions. Section 3.9. If you recall, the number e is important in many instances of exponential growth:. Find the following important limit using graphs and/or tables:. Derivative of . Definition of the derivative!!!. - PowerPoint PPT PresentationTRANSCRIPT
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Section 3.9
DERIVATIVES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS
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If you recall, the number e is important in manyinstances of exponential growth:
1lim 1x
xe
x
Find the following important limit using graphsand/or tables:
0
1limh
h
eh
1
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Derivative of xe
1xe
x xd e edx
0
limx h x
x
h
d e eedx h
0lim
x h x
h
e e eh
0
1limh
x
h
eeh
0
1limh
x
h
eeh
The limit we just figured!
Definition of thederivative!!!
The derivative of this function is itself!!!
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Derivative of xa
ln lnxax x aa e e
lnx x ad da edx dx
ln lnx a de x adx
ln lnx ae a lnxa a
Given a positive base that is not one, we can use a propertyof logarithms to write in terms of :xa xe
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Derivative of ln xlny x
yd de xdx dx
ye x 1y
dydx e
1y dyedx
1dy
dx xImp. Diff. Su
bstit
ution
!
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Derivative of loga xFirst off, how am I able to express in thefollowing way??? lnlog
lnaxxa
1 lnln
d xa dx
lnloglna
d d xxdx dx a
1 1ln a x
COB Formula!
1lnx a
loga x
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Summary of the New Rules(keeping in mind the Chain Rule and any variable restrictions)
u ud due edx dx
lnu ud dua a adx dx
0, 1a a
1lnd duudx u dx
0u
1loglna
d duudx u a dx
0, 1a a
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Now we can realize the FULL POWERof the Power Rule……………observe:
lnn n xx e
ln lnn x de n xdx
lnn n xd dx edx dx
lnn x nex
1nnx
Start by writing x with any real power as a power of e…
1nx nx
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Power Rule for Arbitrary Real Powers
1n nd duu nudx dx
If u is a positive differentiable function of x and n isany real number, then is a differentiable functionof x, and
The power rule works for not only integers, not only rational numbers, but any real numbers!!!
nu
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Quality Practice Problems
3 3y x Find : 3 43 3dy xdx
dydx
34 xy e 312 xdy edx
Find :dydx
4 15 xy 4 14 5 ln 5xdydx
Find :dydx
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Quality Practice Problems
Find :dydx
3lny x 23
1 3dy xdx x
3 , 0xx
1 1ln 5 2
dydx x x
1 , 02 ln 5
xx
5logy xFind :dydx
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Quality Practice Problems
Find :dydx
xy xHow do we differentiate a function when both the base and exponent
contain the variable???
Use Logarithmic Differentiation:1. Take the natural logarithm of both sides of the equation
2. Use the properties of logarithms to simplify the equation
3. Differentiate (sometimes implicitly!) the simplified equation
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Quality Practice Problems
Find :dydx
xy x
ln lnd dy x xdx dx
ln 1dy y xdx
ln 1xdy x xdx
ln lny x x
ln ln xy x1 1 1 lndy x xy dx x
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Quality Practice ProblemsFind using logarithmic differentiation:
dydx
2
2 2
1
xxy
x
2
2 2ln ln
1
xxy
x
21ln ln 2 ln 2 ln 12
y x x x
Differentiate: 2
1 1 1 12 ln 2 22 2 1
dy xy dx x x
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Quality Practice ProblemsFind using logarithmic differentiation:
dydx
2
1 1 1 12 ln 2 22 2 1
dy xy dx x x
2
1 ln 21
dy xydx x x
Substitute: 22
2 2 1 ln 211
xx xx xx
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Quality Practice ProblemsA line with slope m passes through the origin and is tangentto the graph of . What is the value of m? What does the graph look like?
lny x
, lna a1ma
The slope of the curve:
ln 0 ln0
a ama a
The slope of the line:
Now, let’s set them equal…
1ma
lny x
0,0
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Quality Practice ProblemsA line with slope m passes through the origin and is tangentto the graph of . What is the value of m? What does the graph look like?
lny x
, lna a1ma
lny x
ln 1aa a
ln 1a 1a e
So, our slope: 1me
0.368 0,0