Question

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

• A. $P^{\prime \prime \prime }\left(x\right)+P^{\prime }\left(x\right)$
• B. $P\left(x\right)P^{\prime \prime \prime }\left(x\right)$
• C. $P^{\prime \prime }\left(x\right)P^{\prime \prime \prime }\left(x\right)$
• D. a constant

Answer 1

The correct option is B $P\left(x\right)P^{\prime \prime \prime }\left(x\right)$

$y^2=P\left(x\right)$ ..... $\left(i\right)$

Differentiating above equation w.r.t $x$, we get

$2y{y}^{\prime }=P^{\prime }\left(x\right)$

$\Rightarrow$ $2{\left(y^{\prime }\right)}^2+2y{y}^{\prime \prime }=P^{\prime \prime }\left(x\right)$

$\Rightarrow$ $4\left(y^{\prime }\right)y^{\prime \prime }+2y^{\prime }{y}^{\prime \prime }+2y{y}^{\prime \prime \prime }=P^{\prime \prime \prime }\left(x\right)$

Multiply both sides by $P\left(x\right)$

$\Rightarrow$ $y^2[6y^{\prime }{y}^{\prime \prime }+2y{y}^{\prime \prime \prime }]=P\left(x\right)P^{\prime \prime \prime }\left(x\right)$ ..... $\left(ii\right)$

Let $C=2\frac{d}{dx}\left(y^3\frac{d^{2}y}{d{x}^2}\right)$

$C=2[3y^2{y}^{\prime }{y}^{\prime \prime }+y^3{y}^{\prime \prime \prime }]$

$C=P\left(x\right)P^{\prime \prime \prime }\left(x\right)$ ...... From $\left(ii\right)$