Page 289 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes 2. ln y = ln (sec x) tan x = tan x ln sec x,
y' sec x tan x
sec 2 x ln sec x tan x sec 2 x ln sec x tan 2 , x
y sec x
y’ = y(sec x ln sec x + tan x) = (sec x) tan x (sec x ln sec x + tan x).
2
2
2
2
x
2
Example: Find y’ using logarithmic differentiation if y = x /(x – 1) .
Solution:
x x
ln y ln xln x 2ln (x 1),
(x 1) 2
y' x 2 2
ln x ln x 1 ,
y x x 1 x 1
2 x x 2
y' y ln x 1 1 ln x .
x 1 (x 1) 2 x 1

Example: Let 3. f (x ) (x x ) x and g ( ) x x (x x ) .
1. Which of these functions grows more rapidly for sufficiently large x?

2. Differentiate them.
Solution:
x 2
f (x ) (x x x ) x x (x ) ) 2 x x 2 x 2 )
(x 2
(x
1. a. lim g f (x ) ) x lim (x x ) x lim x x x lim x (x x ) x ) x lim x x x 2 1 ( x x 2 ) 0,
(x 2
1 ( x
lim
lim
x
lim
lim
0,
lim
x
a.
x
(x
(x
x g (x ) x x (x ) x ) x x (x ) x ) x x
x x
x
2
because lim x x 2 1 ( x x 2 2 ) . So g grows more rapidly.
because lim x x 1 ( x ) . So g grows more rapidly.
b. Using logarithmi c differenti ation we have :
2. b. Using logarithmi c differenti ation we have :
x
2
x
ln f (x ) ln (x x x ) x x ln x x x ln , x
2
ln f (x ) ln (x ) x ln x x ln , x
f ' (x ) x 2 2
f ' (x ) 2x ln x x x (2 ln x 1),
f (x ) 2x ln x x x (2 ln x 1),
f (x ) x
x 2
x x
f ' (x ) ) f f (x )x (2 ln x x 1) (x x ) x (2 ln x x 1) x x 1 1 (2 ln x x 1),
x 2
(x
x
)x
1) (x
(2 ln
(2 ln
f '
1)
) x
1),
(x
(2 ln
x
x
(x
ln g (x ) ln x (x ) x ) x ln , x
x
ln g (x ) ln x x ln , x
x
ln ln g g (x ) ) ln (x ln ) x ) x x ln x ln ln , x , x
x
ln
ln
ln
ln
(x
x ln
(x ln
x ln
g' (x ) ) x x 1 1 1 1
(x
g'
ln
ln
g g (x ) ln g g (x ) ) ln x x x x x ln x x ln x x 1 1 x ln x x , ,
) ln
x ln
(x
x ln
(x
1
g ( ) x ' g (x )( gln (x )) ln x 1
x ln x
x ln
x (x x ) x x ln x 1 1
x ln x
x 1
x x x ln 2 x ln x .
x
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