Page 290 - DMTH201_Basic Mathematics-1

Unit 10: Logarithmic Differentiation

Notes
Example: Find:

3
d t sin 2 t
.
dt (t 1)(t 2) 2
Solution:

Let:
3
t sin 2 t
y .
(t 1)(t 2) 2
Utilizing logarithmi c differenti ation we get :
3
t sin 2 t
ln
ln y ln 3 t 2 sinln t ln (t 1) 2 ln (t 2),
(t 1)(t 2) 2
1 dy 3 cos t 1 2 3 1 2
y dt t 2 sin t t 1 t 2 2cot t t t 1 t 2 ,
3
d t sin 2 t dy
dt (t 1)(t 2) 2 dt
3 1 2
y 2cot t
t t 1 t 2
3
t sin 2 t 2cot t 3 1 2 .
(t 1)(t 2) 2 t t 1 t 2

Example: Find an equation of the tangent line to the curve:

1 x 1 2 x 1 3 x
y
1 6 x
at x = 0.
Solution:

1 x 1 2x 1 3x (1 ) x 1/2 (1 2 ) x 1/2 (1 3x ) 1/2
y .
1/2
1 x 1 1 2x 3x (1 ) x 1/2 (1 6x ) ) x 1/2 (1 3x ) 1/2
6x 1
(1 2
y 1/2 .
1 6x (1 6x )
Employing logarithmi c differenti ation we obtain :
Employing logarithmi c differenti ation we obtain :
ln y 1 ln 1 ( ) x 1 ln (1 2x ) 1 ln (1 3x ) 1 ln (1 6x ),
2 2 2 2
ln y 1 ln 1 ( ) x 1 ln (1 2x ) 1 ln (1 3x ) 1 ln (1 6x ),
y' 2 1 1 2 3 2 3 , 2
y
y' 2(1 1 ) x 1 1 2x 2(1 3 3x ) 1 3 6x
y 2(1 ) x 1 2x 2(1 3x ) 1 6x ,
y' y 1 1 3 3 .
2(1 ) x 1 2x 2(1 3x ) 1 6x
y' y 1 1 3 3 .
2(1 ) x 1 2x 2(1 3x ) 1 6x
At x 0 we have y 1 0 1 2(0) 1 3(0) 1 6(0) 1. Thus the slope of the tangent line is :
At x 0 we have y 1 0 1 2(0) 1 3(0) 1 6(0) 1. Thus the slope of the tangent line is :
y' (1) 1 1 3 3 0.
x , 0 y 1 1 ( 2 ) 0 1 2(0) 1 ( 2 3 (0)) 1 6(0)
y' (1) 1 1 3 3 0.
x , 0 y 1 1 ( 2 ) 0 1 2(0) 1 ( 2 3 (0)) 1 6(0)
Consequently the equation of the tangent line is y – 1 = 0(x – 0) or y = 1.

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