differentiation of

8/17/2019 Differentiation Of

1/18

DIFFERENTIATIONOF

EXPONENTIALFUNCTIONS


2/18

OBJECTIV

ES:•

apply the properties of exponentialfunctions to simplify dierentiation;

• dierentiate functions involvingexponential functions; and

• solve problems involvingdierentiation of exponentialfunctions.


3/18

The EXPONENTIAL FUNCTION

. ylog x

as writtenbealso maya y function,clogarithmi of inverse the is functionl exponentia the Since

number.real a is x wherea ybydefined is1,a

and 0a a, base with functionl exponentia The

a

x

x

=

=

=≠

>

.


4/18

.

nmanama .1 +=⋅

<

=>−

=

nm if ,m-na

1

nm if , 1

nm if ,nma

na

ma .2

( ) mnanma . =

( ) nbnanab .! =

nb

nan

ba

." =

0a provided ,10

a .# ≠= n1mam

n1anma .$

==

Laws of Exponents

xa .% xlog a=

y x then aaif .& y x

==


5/18

DIFFERENTIATION FORMULA

Der!at!e of Exponenta"F#n$ton The derivative of the exponentialfunction for

any given base and any dierentiablefunction of u.( )

f'x(u where)dx

due (e'

dx

d

f'x(u where)dx

dualna (a'

dx

d

uu

uu

===

==

:ebaseFor

:abasegivenany For


6/18

A. Find the derivative of each of thefollowing natural logarithm and

simplify the result:( )

2 xe x f .1 =

( ) x21e x g .2 −=

( ) x * 12e x! xh . =

( ) ( )

( )2

2

x

x

xe# x+ f

x# e x+ f

=

=

( ) x212

2e x+ g x21

−−

⋅= −

( ) ( ) ( )

+

−= x2e x

1e x! x+ h

x * 1

2

x * 12

EXA%PLE:

( ) x21

x21

x21

e x+ g

x21

−−

•−

−=−

( ) ( ) x21e! x+ h x * 1 +−=

( ) ( )1 x2e! x+ h x * 1 −=

( ) x21

x21e x+ g

x21

−

−−=

−


7/18

2 y x2 x

xye .! +=+

[ ] [ ] [ ] 02 y

+ y x1 y x21 y+ xy xye +−=+⋅+

2 y

+ xy y x2 y

xye+ y

xy xe

−=++

+ xy y2

xy2 xye

y+ y

xye

2 xy −=++

2 xy2 xye y y x

xye2 xy+ y −−=+

+

−−

= xye2 y1 x

xye2 y xy21 y

+ y


8/18

" x!2 x$ y ." +−=

( )

+−+−= " x!2 x

dx

d $ ln" x!

2 x$ + y

( )[ ]! x# $ ln" x!2

x$ + y −+−=

( )( ) " x!2 x$ $ ln2 x2+ y +−−=

( )2

x!ln xh .# =

2 x!

2 x!dxd

( x' + h

=

( ) ( )2 x!

2 xdx

d !ln

2 x!

x+ h =

( ) ( )[ ] x# !ln x+ h =

( ) !ln x# x+ h =

( )2 x!ln xh =

( ) !ln2 x xh =

( ) ( )

= 2 x

dxd !ln x+ h

( ) ( )[ ] x2!ln x+ h =

( ) !ln x# x+ h =


9/18

( ) ( )

++= x2e1 xelog x .#

( ) ( ) ( )elog 1elog x. x2 x +++=

( ) elog e

2eelog

1e

e x+ .

x2

x2

x

x

+⋅

++

=

( ) ( ) ( )( )( )elog

e1e1ee2ee x+ .

x2 x

x x2 x2 x

+++++=

( ) ( )( ) elog ee1ee2e2e

x+ .

x

x2 x

x x2 x2

++

+++

=

( )( )( )

elog ee1e

e2e x+ . x

x2 x

x x2

++++

=


10/18

( )2!

x x"2 x f .$ ⋅=

( ) ( ) ( )!22! x x x x 2dx

d ""

dx

d 2 x+ f +=

( ) ( )[ ] ( )[ ] x x x x x122ln2" x2"ln"2 x+ f !22! +=

( ) [ ]2ln x# "ln"2 x2 x+ f 2 x x2!

+=

( ) ( ) x2ln x# "ln"2 x+ f 2 x1 x2!

+= +

( )2!

x x"2 x f ⋅=

( ) ( )2! x x "2ln x f ln =

( )2!

x x "ln2ln x f ln +=

( ) "ln x2ln x x f ln 2! +=

( )

( ) ( )[ ] ( )[ ] x2"ln x!2ln

x f

x+ f +=

( )

( ) [ ]"ln2ln x# x2

x f

x+ f 2 +=

[ ]"ln2ln x# x2"2 ( x' + f 2 x x2!

+⋅⋅=

( ) ( ) x"ln2ln x# "2 x+ f 2 x1 x 2!

+=

+


11/18

y x" .%! y x +=+

( ) ( ) + y x!+ y"ln"ln y x

+=+( )[ ] ( )ln x!1"ln"+ y x y −=−

( )

( )[ ]1"ln"

ln x!+ y

y

x

−

−=


12/18

A. Find the derivative and simplify the res

( ) 1 x x2

x g .1 +−=

( )22

xln xe x f .2

+=

2ee y . x

x!

+=

( )

x22

2 xlog xh .! ⋅=

( )2 x" x.1 =

2ln y x xe ye.222 y x ++=+

( ) ( )2

/ 1 x x . +=

1 x2

e y.!

1 x2

+=

+

( ) ( ) x2 x2 eeln x f ." −+=

. Apply the appropriate formulas to obtai derivative of the given function and si

EXERCISES:


13/18

Logarithmic Differentiation

!ftentimes" the derivatives of algebraicfunctionswhich appear complicated in form#involving products" $uotients and

powers% can be found $uic&ly by ta&ingthe natural logarithms of both sides andapplying the properties of logarithms

before dierentiation. This method iscalled "o&art'($ )*erentaton.


14/18

'. Ta&e the natural logarithm of bothsides and apply the properties oflogarithms.

(. )ierentiate both sides and reducethe right side to a single fraction.

*. +olve for y, by multiplying the rightside by y.

-. +ubstitute and simplify the result.

Steps n app"+n& "o&art'($ )*ere

ogarithmic dierentiation is also applicablewheneverthe base and its power are both functions.


15/18

x x yif

dx

dy ind .1 =

xln x yln

xln yln x

=

=

ogarithmic dierentiation is also applicablewhenever the base and its power are bothfunctions. #/ariable to variable power.%

0xample:

( ) ( )1 xln1 x

1 x+ y

y

1+=

( ) x x y but y xln1+ y =→+=

( ) ( ) x x xln1+ y +=∴


16/18

( )

( ) ( )1 x2ln1 x yln

1 x2ln yln1 x

+−=

+= −

( ) 1 x1 x2 yif dx

dy ind .2

−+=

( ) ( ) ( )( )11 x2ln21 x2

11 x+ y

y

1++

+

−=

( )( )1 x2ln1 x2

1 x2+ y y

1++

+

−

=

( )( ) ( ) 1- x12x y but y1 x2ln

1 x2

1 x2+ y +=→

++

+−

=

( ) ( ) ( ) ( ) 1- x12x1 x21 x2ln1 x21 x2+ y + + +++−=

( ) ( ) ( )[ ] ( ) 1-1- x12x1 x2ln1 x21 x2+ y ++++−=

( ) ( ) ( )[ ] ( ) 2- x12x1 x2ln1 x21 x2+ y ++++−=∴

( )


17/18

( ) x" x# y . +=

( )" x# ln x yln

" x# ln yln x

+=

+=

++

+

+=

x2

1" x# ln

" x# 2

#

" x#

1 x y+

y

1

( )( ) 12

x

" x# x2

" x# ln" x# x# y+

−+

+++=∴

x2

" x# ln

" x#

x y+

y

1 ++

+=

( )

( )" x# x2

" x# ln" x# x# y+

y

1

+

+++=

( )

( ) ( ) ( ) x" x# y but y

" x# x2

" x# ln" x# x# y+ +=⇒

+

+++=

( )

( ) ( ) x" x#

" x# x2

" x# ln" x# x# y+ +

+

+++=


18/18

( ) 1 x x! y .! −−=

( )

( ) x!ln1 x yln

x!ln yln1 x

−−=

−= −

( ) ( )

−−+

−−

−=1 x2

1 x!ln

x!

11 x+ y

y

1

( )

1 x2

x!ln

x!

1 x+ y

y

1

−

−+

−

−−=

( ) ( ) ( )

( ) 1 x x!2

x!ln x!1 x# + y

y

1

−−

−−+−−=

( ) ( ) ( )

( ) ( ) ( ) 1 x x! y but y

1 x x!2

x!ln x!1 x# y+

−−=⇒

−−

−−+−−=

( ) ( ) ( )

( ) ( ) 1 x x!

1 x x!2

x!ln x!1 x# + y

−−

−−

−−+−−=

( ) ( ) ( )( )

11 x

x!1x2

x!ln x!1 x# y+

−−

−

−

−−+−−=∴

differentiation of

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