Class Meeting#9

Derivatiye

Shprt

Cuts

-These

formulas

are

derived in

calculus

and are

used to

save much

time

in

the

computation

of

derivatives

throughout

the

course.

lf

ffix),

then the

derivative

ofy

with

respect

to x may

be

symbolized

any

one of

the

following

ways:

dvldr f

'(x),

v

1

tlf(x)],

dfldx,

D,y,

D,[f(x)].

(the

flrst 3

are

the most

popular]

Rules

1) If

{r.}:

k,

3 ccnstant,

then,

f

'{x}:

g

2) If

f{x)

:

h(x)

t

g(.x),

then f

'(x):

h'(x)

t

g

'(x).

(for

sums

&

differences}

3) If

{x)

:

[u(x]J',

then f

'(x)

:

n

[u(x)]

"-'u

'(x).

(power

rule)

4)

If'(x)

:

u(x)

v(x),

then

f

'1x;

=

u(.x)

v'(x)

+

v(x)

u '(x).

(product

rule)

5)

If f(x)

=

u{x)

,

then,

f

'(x):

v(x)

u'(x)

*

u(x)

v

'(.x)

(quotient

rule}

v(")

[v(*)]'

Proofl

Recall

that

dyldx

=

lim

Ay

Ax-+0 Ax

Leiy=u/v,whereuandvarefunctionsofx.Considerasmallchangeinx(i.e..*+Ax).Thisproducesachange

in

u and in

v,

since

both

are

functions

of

x. Therefore,

a

change

in

y

is also

produced.

So.

r*piu."

u by

u*Au,

v

by

v*Av, and

y

by

y+Ay.

We

get:

y

+

Ay

=

u_*

Au

Subtract

the

equation

y:

u

and we

get:

Ay:

u*Au

*

u

v*Av

v

v*Av

v

Simplifying

the

right

side, we

get:

Ay

=

v(u+Au)-u(v+Av)

=

v{Au)-u(Av)

v(v+Av)

v(v+try)

Now,

dMde

both sides

by Ax

to

get:

Ay

:

v(ArlAx)-u(AvlAx)

Lx

v(v+Av)

Take the limit

ofboth

sides

as Ax-+G,

we

get:

lim

Ay

:

v(Iim

Au/Ax

)-u

(lim

Av/Ax)

Ax-+0

Ax Ax+0

Ax-+0

v(,;+lim

Av)

Ax+0

Which

gives

us,

dy/dx

=

v duldx

:

u dvldx

v