Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

[email protected] • MTH15_Lec-20_sec_4-3_EXP-n-LOG_Derivatives.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§4.3 Exp & Log

Derivatives



Review §

Any QUESTIONS About• §4.2 → Logarithmic Functions

Any QUESTIONS About HomeWork• §4.2 → HW-19

4.2

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§4.3 Learning Goals

Differentiate exponential and logarithmic functions

Examine applications involving exponential and logarithmic derivatives

Employ logarithmic differentiation

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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



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



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



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



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




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




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




Tangent Line at (0,1) Graphically



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:


x

xdx

d 1ln



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



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



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



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



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



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



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



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



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




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




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




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



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

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All Done for Today

ForPHYS4AStudents

From RigidBody Motion-Mechanics



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22



ConCavity Sign Chart

a b c

−−−−−−++++++ −−−−−− ++++++

x

ConCavityForm

d2f/dx2 Sign

Critical (Break)Points Inflection NO

InflectionInflection



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



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)



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



Prove d(ex)/dx =ex

– D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp. 325-331



Prove d(ex)/dx =ex





Bruce Mayer, PE Chabot College MathematicsProve: xx aaadx

dln

axx

dx

da ln

1log

















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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege