differentiation of
TRANSCRIPT
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8/17/2019 Differentiation Of
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DIFFERENTIATIONOF
EXPONENTIALFUNCTIONS
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8/17/2019 Differentiation Of
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OBJECTIV
ES:•
apply the properties of exponentialfunctions to simplify dierentiation;
• dierentiate functions involvingexponential functions; and
• solve problems involvingdierentiation of exponentialfunctions.
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8/17/2019 Differentiation Of
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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
=
=
=≠
>
.
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8/17/2019 Differentiation Of
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
==
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8/17/2019 Differentiation Of
5/18
DIFFERENTIATION FORMULA
Der!at!e of Exponenta"F#n$ton The derivative of the exponentialfunction for
any given base and any dierentiablefunction of u.( )
f'x(u where)dx
due (e'
dx
d
f'x(u where)dx
dualna (a'
dx
d
uu
uu
===
==
:ebaseFor
:abasegivenany For
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8/17/2019 Differentiation Of
6/18
A. Find the derivative of each of thefollowing natural logarithm and
simplify the result:( )
2 xe x f .1 =
( ) x21e x g .2 −=
( ) x * 12e x! xh . =
( ) ( )
( )2
2
x
x
xe# x+ f
x# e x+ f
=
=
( ) x212
2e x+ g x21
−−
⋅= −
( ) ( ) ( )
+
−= x2e x
1e x! x+ h
x * 1
2
x * 12
EXA%PLE:
( ) x21
x21
x21
e x+ g
x21
−−
•−
−=−
( ) ( ) x21e! x+ h x * 1 +−=
( ) ( )1 x2e! x+ h x * 1 −=
( ) x21
x21e x+ g
x21
−
−−=
−
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8/17/2019 Differentiation Of
7/18
2 y x2 x
xye .! +=+
[ ] [ ] [ ] 02 y
+ y x1 y x21 y+ xy xye +−=+⋅+
2 y
+ xy y x2 y
xye+ y
xy xe
−=++
+ xy y2
xy2 xye
y+ y
xye
2 xy −=++
2 xy2 xye y y x
xye2 xy+ y −−=+
+
−−
= xye2 y1 x
xye2 y xy21 y
+ y
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8/17/2019 Differentiation Of
8/18
" x!2 x$ 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 =
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8/17/2019 Differentiation Of
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
++++
=
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8/17/2019 Differentiation Of
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!
+=
+
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8/17/2019 Differentiation Of
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
−
−=
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8/17/2019 Differentiation Of
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:
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8/17/2019 Differentiation Of
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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.
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8/17/2019 Differentiation Of
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'. 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.
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8/17/2019 Differentiation Of
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 +=∴
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8/17/2019 Differentiation Of
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( )
( ) ( )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 ++++−=∴
( )
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8/17/2019 Differentiation Of
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+ +
+
+++=
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8/17/2019 Differentiation Of
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+
−−
−
−
−−+−−=∴