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  4. If y2=p(x) is a polynomial of degree 3 , then 2(d/d x)(y3 (d2 y/d x2)) is equal to :-

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

NTA AbhyasNTA Abhyas 2022

Solution:

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