Page 286 - DMTH201_Basic Mathematics-1

Unit 10: Logarithmic Differentiation

Multiply both sides of this equation by y, getting Notes
2lnx 3x 2 tanx 3 ln(sec )
x
x
y = y
x
x
3 2ln x 3x 2 tan x 3 ln(sec )
x
= x ln x (sec ) x
x
x
(Combine the powers of x.)
= x (ln x 1) (sec ) x 3x 2ln x 3x 2 tan x 3 ln(sec ) x
x
Logarithmic Differentiation

df x
x
:
Example: Determine of f x cos( )
dx
x
x
f ( ) = cos( )
x
x
ln f ( ) = ln cos( )x
x
x
ln f ( ) = x ln cos( )
x
d d
ln f ( ) = x ln f ( )
x
x
dx dx
1 df 1
x
. = ln cos( )x x sin( )
x
f dx cos( )
1 df
x
. = ln cos( )x x tan( )
f dx
df
x
x
= f ln cos( ) x tan( )
dx
df x x
x
x
x
x
= cos( ) ln cos( ) x cos( ) tan( )
dx
And thus
df x x
x
x
= cos ( )ln cos( )x x x cos ( )tan( )
dx
Example: Differentiate y = (2x) sin x .
Solution:
Alternate 1
y = (2x) sin x = e sin x ln 2x ,

y ' e sin x ln 2 x cos x ln 2 x ( sin x ) 2 ( 2 x sin x cos x ln 2 x sin x .

)
2 x x

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