If $y^{2} = P(x)$ is a polynomial of degree 3, then $2\frac{d}{dx}\left\lbrack y^{3}\frac{d^{2}y}{dx^{2}} \right\rbrack$ equals

Question

Dr. Abubacker · Accepted Answer

(c)
$y^{2} = P(x) \Rightarrow 2yy^{'} = P^{'}(x)$ …(i)
$\Rightarrow (2y)y^{''} + y^{'}\left( 2y^{'} ight) = P^{''}(x)$
$\Rightarrow 2yy^{''} = P^{''}(x) - 2\left( y^{'} ight)^{2}$
$\Rightarrow 2y^{3}y^{''} = y^{2}P^{''}(x) - 2\left( yy^{'} ight)^{2}$
$= y^{2}P^{''}(x) - 2\frac{\left\{ P^{'}(x) ight\}^{2}}{4}$
[from Eq.(i)]
$\Rightarrow 2y^{3}.y^{''} = P(x)P^{''}(x) - \frac{1}{2}\left\{ P^{'}(x) ight\}^{2}$
$	herefore\frac{d}{dx}\left( 2y^{3}.y^{''} ight)$
$= P(x)P^{'''}(x) + P^{''}(x)P^{'}(x) - P^{'}(x)P^{''}(x)brack$
$= P(x).P^{'''}(x)$
$\Rightarrow 2\frac{d}{dx}\left( y^{3}\frac{d^{2}y}{dx^{2}} ight) = P(x)P^{'''}(x)$

Q. If $y^{2} = P(x)$ is a polynomial of degree 3, then $2\frac{d}{dx}\left\lbrack y^{3}\frac{d^{2}y}{dx^{2}} \right\rbrack$ equals

Explanation

Reviewed by Dr. Abubacker

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