section 3.9 derivatives of exponential and logarithmic functions
TRANSCRIPT
If you recall, the number e is important in manyinstances of exponential growth:
1lim 1
x
xe
x
Find the following important limit using graphsand/or tables:
0
1lim
h
h
e
h
1
Derivative of xe
1xe
x xde e
dx
0
limx h x
x
h
d e ee
dx h
0lim
x h x
h
e e e
h
0
1lim
hx
h
ee
h
0
1lim
hx
h
ee
h
The limit we just figured!
Definition of thederivative!!!
The derivative of this function is itself!!!
Derivative of xa
ln lnxax x aa e e
lnx x ad da e
dx dx ln lnx a d
e 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
Derivative of ln x
lny x
yd de x
dx dx
ye x 1y
dy
dx e
1y dye
dx
1dy
dx xImp. Diff. Su
bstit
ution
!
Derivative of loga x
First off, how am I able to express in thefollowing way???
lnlog
lna
xx
a
1ln
ln
dx
a dx
lnlog
lna
d d xx
dx dx a
1 1
ln a x
COB Formula!
1
lnx a
loga x
Summary of the New Rules(keeping in mind the Chain Rule and any variable restrictions)
u ud due e
dx dx lnu ud du
a a adx dx
0, 1a a
1ln
d duu
dx u dx
0u
1log
lna
d duu
dx u a dx
0, 1a a
Now we can realize the FULL POWERof the Power Rule……………observe:
lnn n xx e
ln lnn x de n x
dx lnn n xd d
x edx dx
lnn x ne
x 1nnx
Start by writing x with any real power as a power of e…
1nx nx
Power Rule for Arbitrary Real Powers
1n nd duu nu
dx 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
Quality Practice Problems
3 3y x Find : 3 43 3dy
xdx
dy
dx
34 xy e 312 xdye
dxFind :
dy
dx
4 15 xy 4 14 5 ln 5xdy
dx Find :
dy
dx
Quality Practice Problems
Find :dy
dx 3lny x 2
3
13
dyx
dx x
3, 0x
x
1 1
ln 5 2
dy
dx x x
1, 0
2 ln 5x
x
5logy xFind :dy
dx
Quality Practice Problems
Find :dy
dxxy x
How 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
Quality Practice Problems
Find :dy
dxxy x
ln lnd d
y x xdx dx
ln 1dy
y xdx
ln 1xdyx x
dx
ln lny x x
ln ln xy x1 1
1 lndy
x xy dx x
Quality Practice ProblemsFind using logarithmic differentiation:
dy
dx
2
2 2
1
xxy
x
2
2 2ln ln
1
xxy
x
21ln ln 2 ln 2 ln 1
2y x x x
Differentiate: 2
1 1 1 12 ln 2 2
2 2 1
dyx
y dx x x
Quality Practice ProblemsFind using logarithmic differentiation:
dy
dx
2
1 1 1 12 ln 2 2
2 2 1
dyx
y dx x x
2
1ln 2
1
dy xy
dx x x
Substitute:
22
2 2 1ln 2
11
xx x
x xx
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 a1
ma
The slope of the curve:
ln 0 ln
0
a am
a a
The slope of the line:
Now, let’s set them equal…
1m
a
lny x
0,0