-To

calculate

the

derivative

function,

we

use

Newton's

Difference

Quotient

definition,

as

follows:

f

'

(x)

=

lim

f(.x+h)-f(,x)

h-*0

h

The

Zod

form

is

useful

for

proving

many

short-cuts

for

finding

derivatives'

1)

The

geometric

meaning

(slope

of

the

tangent

line

to

the

function

at

a

value

of

x)'

remainJ

the

same

as

illustrated

by

the

sketches

below.

gt*)

F

!*(,

*/v*l)

"j*n*''

gc'c)

y+

F-

Note:

It can

also

be

stated

as:

dy =

lim

Ay

dx

Ax-O

Ax

r

t

2)

For distance-time

functions,

s:f(t),

where

s is

distance

and

t

is time'

the

Derivative would

be

ds

:

lim

As

dt

At+o

at

This

gives

the

velocity

of

the

particle

at

any

gwen time

t

3)

The

most

general

interpretation

of

the

derivative

glves the

rate

o.f

:l*g.

(instantaneous)

of

the

dependent

variable

with

respect

to

(wrt)

{r9

indenendent

variable

at

a

given

value

of the

irra.prnJrnt

variabl..''wh*,

iate

of

chang;

is

mentioned,

it

will

be

assumed

to

be

instantaneous.

For

example,

drldw

would

give the

rate

of

change

of

the

quantlty

r

wrt

w

at

some

w

value'

If

R(x)

is aRevenue

function

when

x

items

are

sold,

dR/dx

would

give how

fast

Revenue

Is

changing at

a

given

level

of

sales

(when x

items

are

sold).

If

v(x) gives

the

volume

of a

sphere

(ba11) witha-radius

ofx,

dv/dx

would

give

how

fast

the volume is

changing

at

a

given

length

or*.

mi*

of

this

as

an

expanding

sphere'

so, interpretation

3

opens

up

many

avenues

for

solving

many

important

dynamic

problems

4) For

business

applications,

the

derivative

gives

Marginal

values

for

Revenue'

Cost'

& Profit at

a

given

level

of

sales

or

production'