Page 310 - DMTH201_Basic Mathematics-1

Unit 12: Successive Differentiation

5. Let y xe x Notes
dy x x
xe e 1
dx

2
d y x x
x 1 e e 1
dx 2
x 2 e x

d 3 y
(x ) 2 e x e x 1 .
dx 3
x ( e ) 3 x

d 4 y x x
(x ) 3 e e 1 .
dx 4
x ( e ) 4 x

2
d y
Example: If x 2 xy y 2 0, prove that 0 .
dx 2
Solution: x 2 xy y 2 0

Differentiating w.r.t. x, we get
dy dy
2x x y 2y 0
dx dx
dy
x 2y 2x y
dx

dy 2x y
dx x 2y

dy dy
x 2y 2 2x y 1 2
2
d y dx dx
dx 2 x 2y 2

2x y (2x y )
y
(x 2 ) 2 (2x y ) 1 2
x 2y (x 2 )
y
y
(x 2 ) 2
x 2y 3y 2x y 3x
3
x 2y

3xy 6y 2 6x 2 3xy
3
x 2y

LOVELY PROFESSIONAL UNIVERSITY 303