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Differentiation (p-1) — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Differentiation (p-1)2 Marks
Question
Find $\frac{d^2 y}{d x^2}$, if $y = x ^2 \cdot e ^{ x }$
✓

Answer

$
y=x^2 \cdot e^x
$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{d y}{d x} & =\frac{d}{d x}\left(x^2 e^x\right)=x^2 \frac{d}{d x}\left(e^x\right)+e^x \cdot \frac{d}{d x}\left(x^2\right) \\
& =x^2 \cdot e^x+e^x \times 2 x=e^x\left(x^2+2 x\right)
\end{aligned}
$
Differentiating again w.r.t. $x$, we get
$
\begin{aligned}
& \begin{aligned}
\frac{d^2 y}{d x^2} & =\frac{d}{d x}\left[e^x\left(x^2+2 x\right)\right] \\
& =e^x \cdot \frac{d}{d x}\left(x^2+2 x\right)+\left(x^2+2 x\right) \cdot \frac{d}{d x}\left(e^x\right) \\
& =e^x(2 x+2)+\left(x^2+2 x\right) \cdot e^x
\end{aligned} \\
& = e ^{ x }\left(2 x +2+ x ^2+2 x \right) \\
& = e ^{ x }\left( x ^2+4 x +2\right) .
\end{aligned}
$

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