V

Derivatiys

of the f,xoonential

&

Ioqarithmic

functions

-Recstl

that

e

=

lim

(I

+

lln)"

fl-'€

The

Natu.ra!

lo$

fu.nction:

f(x)

=

ln(x)

*+

f

'1x1

=

Iim

fl"xtlf):f[x)

=

lim

lrt(.xth)/xl

=

lim

(1/h]n(1+hlx)

h*+0hh-+0hh-+0

L,et

n

:

x/h

for

any

x in

the

domain, So, a$ h-+0, n-+m.

Then, lim

(dx)ln(l+1in)

:

(l/x)lim

ln(l+l/n)"

il*+ao

n*)s

=

(llxlne

:

I/x. So,

f

'(x)

=

l/x

Ch+in

Fplp

for

der,F,qttves

-@gslgru)

(see

rny website): dyldx

=

(dy/duXdddx)

-Ire-fxponentiatnrgction;

y:e*-+lny=x

>d&y)=d

(x)

-+d(lny)&

:

I

-r(l/y[y:l

-tdy=y=e*

dx

dx dy

dx

dx dx

-Ngte:

The

process

oftaking

the

ln

ofboth sides and

then differentiating is

called

Logarithmic-Differentiation.

The Gener,al

ExoonentiflI

functi.on

-Formi

y:

a*

,

where u

is

a function ofx.

Let's

find

dyldx

by

log

differentiation.

y:a'-+

lny

=

ulna

-+

d

(lny)

=

$na)

(drldx)

-+ (lly)

dyldx:

(lna) (drldx)

+dy/dr:

y(lna)du/dx

=f,u

lnn

duldx

dx

Exsmptes:

(give

a

yariety

of

different tpes)