If `y^(2) = ax^(2) + bx + c ` , where a,b,c, are constants , then
` y^(3) (d^(2) y)/(dx^(2))` is equal to

Options

A. a constant
B. a function of x
C. a function of x
D. a function of x and y both

Answer

A

Explanation

Given , ` y^(2) = ax^(2) + bx + c `
On differentiating both sides w.r.t.x, we get
` 2y (dy)/(dx) = 2 ax + b `
Again , differentiating w.r.t.x, we get
`2((dy)/(dx))^(2) + 2y (d^(2) y)/(dx^(2)) = 2 a rArr y (d^(2) y)/(dx^(2) ) = a- ((dy)/(dx))^(2)`
` rArr y (d^(2)y)/(dx^(2) ) = a- ((2ax + b)/(2y))^(2)`
`rArr y (d^(2) y)/(dx^(2)) = (4xy^(2) - (2ax + b)^(2))/(4y^(2))`
` rArr 4y^(3) (d^(2)y)/(dx^(2) )= 4a (ax^(2) + bx + c)-(4a^(2)x^(2) + 4abx + b^(2))`
`rArr 4y^(3) (d^(2) y)/(dx^(2) )= 4ac - b^(2)`
` rArr y^(3) (d^(2) y)/(dx^(2)) = (4ac - b^(2))/(4) ` = constant