If `y^2 = P(x)` is polynomial of degree 3, then `2(d/dx)(y^(3).(d^2y)/dx^2)` is equal to

Options

A. p(x)p"(x)
B. p"(x)p"(x)
C. p(x)p"(x)
D. A constant

Answer

C

Explanation

Given that ` y^(2) =p (x)`
Differentiating
` Rightaarrow 2yy_(1) = p'(x)`
` Rightarrow 2y_(1) = ( p'(x))/(y)`
Differentiatiing again ,
`Rightarrow 2y_(2) = (yp"(x)-p(x)y_(1))/(y^(2)),[y_(2)= (d^(2)y)/(dx^(2))]`
`Rightarrow2y_(2)=(yp(x)-(p'(x).p(x))/(2y))/y^(2)`
`= ((2y^(2)p'(x)-p(x))^(2))/(2y^(3))`
`Rightarrow2y^(3)y_(2)=1/2[2y^(2)p'(x)-(p'(x))^(2)]`
`Rightarrow2y^(3)y^(2)=1/2[2p(x)p'(x)-(p(x))^(2)]`
` Rightarrow 2d/(dx) (y^(3) y_(2))`
`1/2[2p '(x)p(x) +2p (x) p(x)-2p (x) p(x) ]`
= p (x) p (x)`