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Chabot Mathematics. §4.3 Exp & Log Derivatives. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. 4.2. Review §. Any QUESTIONS About §4.2 → Logarithmic Functions Any QUESTIONS About HomeWork §4.2 → HW-19. §4.3 Learning Goals. - PowerPoint PPT Presentation
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BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
§4.3 Exp & Log
Derivatives
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 2
Bruce Mayer, PE Chabot College Mathematics
Review §
Any QUESTIONS About• §4.2 → Logarithmic Functions
Any QUESTIONS About HomeWork• §4.2 → HW-19
4.2
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 3
Bruce Mayer, PE Chabot College Mathematics
§4.3 Learning Goals
Differentiate exponential and logarithmic functions
Examine applications involving exponential and logarithmic derivatives
Employ logarithmic differentiation
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 4
Bruce Mayer, PE Chabot College Mathematics
Derivative of ex
For any Real Number, x
Thus the ex fcn has the unusual property that the derivative of the fcn is the ORIGINAL fcn• The proof of this is quite complicated. For
our purposes we treat this as a formula– For a good proof (in Appendix) see:
D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp. 325-331
xx eedx
d
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 5
Bruce Mayer, PE Chabot College Mathematics
Derivative of ex
Using the “repeating” nature of d(ex)/dx
Meaning of Above: for any x-value, say x = 1.9, All of these y-related quantities are equal at e1.9 = 6.686• The y CoOrd:• The Slope:• The ConCavity:
xxxxx eedx
d
dx
dyee
dx
dyey
'''
6.686 1.9,
686.69.1 xdxdy
686.69.122 xdxyd
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 6
Bruce Mayer, PE Chabot College Mathematics
Example ex Derivative
Differentiate: Using Rules
• Product• Power• ex
xey x 1
xedx
dy
dx
d x 1
xexedx
dy xx 1)1(2
1 2/1
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 7
Bruce Mayer, PE Chabot College Mathematics
Chain Rule for eu(x)
If u(x) is a differentiable function of x then
Using the ex derivative property
dx
xdue
xdu
de
dx
d xuxu
dx
xdue
dx
xdue
xdu
d xuxu
xuedx
duee
dx
d uuu 'Or
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 8
Bruce Mayer, PE Chabot College Mathematics
Example Tangent Line
Find the equation of the tangent line at x = 0 for the function:
SOLUTION: Use the Point-Slope Line Eqn,
y-yAP = m(x-xAP), with• Anchor Point, (xAP,yAP):
• Slope at the Anchor Point:
xxexf 2
1 ,0 ,00 ,0 002
ef
00 xdxdfm
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 9
Bruce Mayer, PE Chabot College Mathematics
Example Tangent Line
Find Slope at x = 0
Let: Then:
Thus:
And by Chain Rule
12and 2 xxxdx
du
dx
dee
du
d uu
xxu 2
uxx ee 2
12122
2
xexem
dx
du
du
ede
dx
dm
xxu
uxx
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 10
Bruce Mayer, PE Chabot College Mathematics
Example Tangent Line
Then m at x = 0
Using m and the Anchor-Point in the Pt-Slope Eqn
Convert Line-Eqnto Slope-Intercept form
11110200
0
22
eedx
dm
x
xx
APAP xxmyy
1 xy
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 11
Bruce Mayer, PE Chabot College Mathematics
Example Tangent Line
Tangent Line at (0,1) Graphically
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 12
Bruce Mayer, PE Chabot College Mathematics
Derivative of ln(x) = loge(x)
For any POSITIVEReal Number, x
Thus the ln(x) fcn has the unusual property that derivative Does NOT produce another Log• The proof of this is quite complicated. For
our purposes we treat this as a formula– For a good proof (in Appendix) see:
D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp. 325-331
x
xdx
d 1ln
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 13
Bruce Mayer, PE Chabot College Mathematics
Example ln Derivative
Find the Derivative of: Using Rules
• Quotient• Power• ln(x)
x
x
dx
d
dx
dy
ln1
2
2ln1
ln2
x
xxx
xxxfy ln12
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 14
Bruce Mayer, PE Chabot College Mathematics
Chain Rule for ln(u(x))
If u(x)> 0 is a differentiable function of x then
Using the ln(x) derivative property
dx
xduxu
xdu
dxu
dx
d lnln
dx
xdu
xudx
xduxu
xdu
d
1ln
xuudx
du
uu
dx
d'
11lnOr
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 15
Bruce Mayer, PE Chabot College Mathematics
Derivative of ax & loga(x)
For Base a with a>0 and a≠1, then for ALL x:
For Base a with a>0 and a≠1, then for ALL x>0:
Prove Both on White/Black Board
xx aaadx
dln
ax
xdx
da ln
1log
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 16
Bruce Mayer, PE Chabot College Mathematics
Example Revenue RoC
The total number of hits (in thousands) to a website t months after the beginning of 1996 is modeled by
The Model for the weekly advertising revenue in ¢ per hit:
Use the Math Models to determine the daily revenue change at the beginning of the year 2005
4ln200 ttH
ttr 1.025
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 17
Bruce Mayer, PE Chabot College Mathematics
Example Revenue RoC
SOLUTION: The rate of change in Total Revenue,
R(t), is the Derivative of the Product of revenue per hit and total hits:
)1.025()4ln(200 ttdt
dtrtH
dt
dtR
dt
d
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 18
Bruce Mayer, PE Chabot College Mathematics
Example Revenue RoC
Thus
Next find t in months for 1996→2005
Then the rate derivative at t = 108 mon
A units analysis
)4ln(204
205000'
tt
ttR
dt
tdR
mon 108yr 1
mon 12yrs 19962005 t
)4108ln(204108
)108(205000
108
dt
dR
mon
kCent
mon
HitCentkHits
t
rH
t
R
dt
dR
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 19
Bruce Mayer, PE Chabot College Mathematics
Example Revenue RoC
The units on H are kHits, and units on r are ¢/Hit. The units on time were months so the derivative has units k¢/mon. Convert to $/mon:
STATE: at the beginning of 2005 the website was making about $690.13 LESS each month that passed.
mon
$ 13.690
kCent
$ 10
mon
kCent 013.69
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 20
Bruce Mayer, PE Chabot College Mathematics
Helpful Hint Log Diff
Logarithmic Differentiation Some derivatives are easier to calculate
by • first take the natural logarithm of the
expression• Next judiciously use the log rules• then take the derivative of both sides of the
equation• finally solve for the derivative term
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 21
Bruce Mayer, PE Chabot College Mathematics
Example Using Log Diff
Using logarithmic differentiation to find the df/dx for:
SOLUTION: Computing the derivative directly would
involve the repeated use of the product rule (not impossible, but very tedious)
Instead, use properties of logarithms to first expand the expression
3 223 13)( xexxf x
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 22
Bruce Mayer, PE Chabot College Mathematics
Example Using Log Diff
Let y = f(x) → Then take the natural logarithm of both
sides:
Use the Power & Log Rules
Now Take the Derivative of Both Sides
3 223 13 xexyxf x
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 23
Bruce Mayer, PE Chabot College Mathematics
Example Using Log Diff
By the Chain Rule
Then
Or
• This is a form of Implicit Differentiation; Need to algebraically Isolate dy/dx
xxxdx
dy
y2
1
1
3
11
312
dx
dy
ydx
dyy
dy
dy
dx
dy
dy
dy
dx
d
1lnlnln
1ln
3
12ln33ln
1ln 2xxx
dx
d
dx
dy
yy
dx
d
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 24
Bruce Mayer, PE Chabot College Mathematics
Example Using Log Diff
Solving for dy/dx
Recall
Thus
• This result would have much more difficult to obtain without the use of the Log transform and implicit differentiation
33
21
32x
x
xy
dx
dy
33
21
313'
23 223
x
x
xxex
dx
xdfxf x
xxxdx
dy
y2
1
1
3
11
312
dx
xdf
dx
dyxfyxexy x and13 3 223
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 25
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard Work
Problems From §4.3• P76 → Per Capita Growth• P90 → Newtons Law of (convective)
Cooling– Requires a Biot Number* of Less than 0.1
Bi → INternal Thermal ResistanceEXternal Thermal Resistance
*B. V. Karlekar, R. M. Desmond, Engineering Heat Transfer, St. Paul,
MN, West Publishing Co., 1977, pp. 103-110
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 26
Bruce Mayer, PE Chabot College Mathematics
All Done for Today
ForPHYS4AStudents
From RigidBody Motion-Mechanics
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 27
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
–
srsrsr 22
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 28
Bruce Mayer, PE Chabot College Mathematics
ConCavity Sign Chart
a b c
−−−−−−++++++ −−−−−− ++++++
x
ConCavityForm
d2f/dx2 Sign
Critical (Break)Points Inflection NO
InflectionInflection
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 29
Bruce Mayer, PE Chabot College Mathematics
Summary of Log Rules
For any positive numbers M, N, and a with a ≠ 1, and whole number p
log log log ;a a aM
M NN
log log ;pa aM p M
log .ka a k
log ( ) log log ;a a aMN M N Product Rule
Power Rule
Quotient Rule
Base-to-Power Rule
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 30
Bruce Mayer, PE Chabot College Mathematics
Change of Base Rule
Let a, b, and c be positive real numbers with a ≠ 1 and b ≠ 1. Then logbx can be converted to a different base as follows:
logb x loga x
loga b
log x
logb
ln x
lnb
(base a) (base 10) (base e)
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 31
Bruce Mayer, PE Chabot College Mathematics
Derive Change of Base Rule
Any number >1 can be used for b, but since most calculators have ln and log functions we usually change between base-e and base-10
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 32
Bruce Mayer, PE Chabot College Mathematics
Prove d(ex)/dx =ex
– D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp. 325-331
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 33
Bruce Mayer, PE Chabot College Mathematics
Prove d(ex)/dx =ex
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 34
Bruce Mayer, PE Chabot College Mathematics
D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp. 325-331
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 35
Bruce Mayer, PE Chabot College MathematicsProve: xx aaadx
dln
axx
dx
da ln
1log
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 36
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 37
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 38
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 39
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 40
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 41
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 42
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 43
Bruce Mayer, PE Chabot College Mathematics