Class

meeting

#5 addition

Example:

find f

,(x)

for

(x)

=

3

-2x_

4x2 using

the definition

of f

,(x)

We need

to complete

a

4-step

process

ending up

with

ffO

ry

This

4-step

process

is called

the delta

process, if

we

use

Ax

instead of

h.

Start

with

f(x+h):

3

-

2(x+h)

-

4(x+h)'?

*

3

-2x

-2h

-4(x'+zxh+ h')

:3-2x-Zh-4x2

-8xh-4h2

Subtract

(x)

from above:

(x+h)

-

f(x):

-2h-8xh-4h'?

(all

terms

remaining must

contain h)

Divide above

by

h:

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

*

-2-8x

4h

(you

will

lose a

power

of

h from

each term)

h

finally,

take

the limit as

h

--*

0

(only

terms

affected

are those that

contain h)

lim f(x+h)-ffu)

=

-2-8x -4(0)

=

-2-8x

(using

the

substitution

method for limits)

h+0 h

The

left side

is f

'(x)

by

definition,

so

f

'(x):

-2 -8x

This

is the

general

formula

for the derivative.

To find the

derivative

at a

specified

point,

just

substitute

ttre x-value

of

the

point

into the expression.

(i.e.,

f

'(0):

-2 means

the slope

of the tangent

at x=0

is

-2).

((0)

:3

gives

the

y-value at x:0)