Page 312 - DMTH201_Basic Mathematics-1

Unit 12: Successive Differentiation

Notes
m 1 1
m x x 2 1 1 2x
2 x 2 1
x 2 1 x
m 1
m x x 2 1
x 2 1
m
m x 2 1 x
i.e., y 1
x 2 1

x 2 1y 1 my ...(i)

Differentiating again w.r.t. x, we get
1
x 2 1y 2 y 1 2x my 1
2 x 2 1
Multiplying throughout by x 2 1 , we get

x 2 1 y 2 xy 1 my 1 x 2 1

x 2 1 y 2 xy 1 m my (using (1))

2
i.e., x 2 1 y 2 xy 1 m y
2
x 2 1 y 2 xy 1 m y 0
Alternate: Squaring equation (i)

2
We get x 2 1 y 2 1 m y 2
Differentiating w.r.t. x,
x 2 1 2y y 2 y 2 1 2x m 2 2yy 1 (cancelling 2y )
1
1
2
x 2 1 y 2 xy 1 m y
2
i.e., x 2 1 y 2 xy 1 m y 0

2
Example: If y ax n 1 bx n , prove that x y n n 1 y 0 .
2
Solution: y ax n 1 bx n
)
y 1 (n 1)ax n 1 1 ( b n x n 1
n 1 ax n bnx n 1
y 2 n 1 nax n 1 bn n 1 x n 1 1
y n 1 nax n 1 bn n 1 x n 2
2
x 2 y (n ) 1 nax n 1 . x 2 bn (n ) 1 x n 2 .x 2
2
2
i.e., x y 2 n 1 nax n 1 bn n 1 x n

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