Find the second order derivative of the following : x^2 e^x - Vidyadip

Question

Find the second order derivative of the following : $x^2 e^x$

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Answer

Let $y=x^2 e^x$
Differentiate w.r.t. $x$
$
\begin{aligned}
& \frac{d y}{d x}=\frac{d}{d x}\left(x^2 e^x\right) \\
& \frac{d y}{d x}=x^2 \frac{d}{d x}\left(e^x\right)+e^x \frac{d}{d x}\left(x^2\right) \\
& \frac{d y}{d x}=x^2 e^x+2 x e^x=e^x\left(x^2+2 x\right)
\end{aligned}
$
Differentiate w.r. $t . x$
$
\begin{aligned}
& \frac{d}{d x}\left(\frac{d y}{d x}\right)=\frac{d}{d x}\left[e^x\left(x^2+2 x\right)\right] \\
& \frac{d^2 y}{d x^2}=e^x \frac{d}{d x}\left(x^2+2 x\right)+\left(x^2+2 x\right) \frac{d}{d x}\left(e^x\right) \\
& =e^x(2 x+2)+\left(x^2+2 x\right)\left(e^r\right) \\
& =\left(x^2+4 x+2\right) e^x \\
& \frac{d^2 y}{d x^2}=\left(x^2+4 x+2\right) e^x \\
\end{aligned}
$

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