If `y^2=a x^2+b x+c` , then `y^3(d^2y)/(dx^2)` is (a) a constant (b) a function of `x` only (c) a function of `y` only (d) a function of `x` and `y`

Options

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

Explanation

`y^(2)=ax^(2)+bx+c`
`"or "2y(dy)/(dx)=2ax+b`
`"or "2((dy)/(dx))^(2)+2y(d^(2)y)/(dx^(2))=2a`
`"or "y(d^(2)y)/dx^(2)=a-((dy)/(dx))^(2)`
`=a-((2ax+b)/2y)^(2)`
`=(4ay^(2)-(2ax+b)^(2))/(4y^(2))`
`"or "4y^(3)(d^(2)y)/(dx^(2))=4a(ax^(2)+bx+c)-(4a^(2)x^(2)+4bax+b^(2))`
`"or "4y^(3)(d^(2)y)/(dx^(2))=4ac-b^(2)= constant`