Question

If $y^2=p\left(x\right)$ is a polynomial of degree $3$ , then $2\frac{d}{dx}\left(y^3\frac{d^{2}y}{d{x}^2}\right)$ is equal to :-

• A. $p^{\prime \prime \prime }(x)+p^{\prime }(x)$
• B. $p^{\prime \prime }(x)+p^{\prime \prime \prime }(x)$
• C. $p^{(x)}{p}^{\prime \prime \prime }(x)$
• D. None of these

Answer 1

Answer: (C)

$p^{\prime }(x)=2y{y}^{\prime }$
$p^{\prime \prime }(x)=2y{y}^{\prime \prime }+2y^2$
$p^{\prime \prime \prime }(x)=2y{y}^{\prime \prime \prime }+6y^{\prime }{y}^{\prime \prime }$
Also $2\frac{d}{dx}\left(y^3\frac{d^{2}y}{d{x}^2}\right)=2\frac{d}{dx}\left(y^3{y}^{\prime \prime }\right)$
$=2\left(y^3{y}^{\prime \prime \prime }+3y^2{y}^{\prime }{y}^{\prime \prime }\right)$
$=y^2\left(2y{y}^{\prime \prime \prime }+6y^{\prime }{y}^{\prime \prime }\right)=p(x{)}^{\prime \prime \prime }(x)$