Example

probler.n:

rf (p)=10,000e*"oRepre$ents

a

demand

function

for

a

product,

where

p

is

the

price

in

dollars,

fud

(2)

andf

,(Z)

and

interpret

these

results.

Example

problem:

How

fast

is

the

volume

of

an

expanding

cube

increasing

with

respect

to

its

total

surface

area

when

its

edge

is

one

inch

long?

Two

inches

long?

If

the

trrfu""

*iin.i"ures

at

a

constant

rate,

does

the

volume

increase

at

a

constant

rate?

lmplicit

Differentiaion

-This

technique

is

used

when

functions

are.given

imqlcitly

(occurs

frequently

in

applications).

what

this

meansis

that

the

dependent

variable

is

not

isolated

on

oo"

rid"

oirirc

equation

(not

solved

for).

Example:

Take

x2

*

y',

:l-

This

equation

defines

a circle

,

center

at

(0,0),

radius

of

I,

Find

dyldx.

You

need

to

take

the

derivative

of both

sides

ofthe

equation

wrt

x.

Do.

Logarithmic

Differeutiation

-This

method

is

used

for

very

complicated

functions

involved

with

products,

quotients,

&

powers

and

where

basic

formulas

can

not

be

used.

(like

trre

rouowing

;;;i;i

ffirT''

we have

no

formula

for

this

one'

First

rake

the

narural

log (ln)

of

both

sides

&

simplify

lny

=

1t1**

)

-+

Inv:

x lnx

(now

take

the

derivative

of

both

sides

irnplicitly)

-+

(l/y)(dyldx)

:

x(l/x)+(lnx)(1)

-+dyldx

:

x'

(l+lnx),

p0.

Example:

Do

{x)

=

(ro'-2Xx3-3Xx4 _4)