Find $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$, where $\mathrm{y}=\log \left(\frac{\mathrm{x}^{2}}{\mathrm{e}^{2}}\right)$
Formula: -
(i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$
(ii) $\frac{\mathrm{d}\left(\mathrm{e}^{\mathrm{ax}}\right)}{\mathrm{dx}}=\mathrm{ae}^{\mathrm{ax}}$
(iii) $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{x}^{\mathrm{n}}=\mathrm{n} \mathrm{x}^{\mathrm{n}-1}$
Given: -
$y=\log \left(\frac{x^{2}}{e^{2}}\right)$
Differentiating w.r.t $x$
$\frac{d y}{d x}=\frac{1}{x^{2}} \cdot \frac{1}{e^{2}} 2 x=\frac{2}{x}$
Again Differentiating w.r.t $x$
$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=2\left(-\frac{1}{\mathrm{x}^{2}}\right)=-\frac{2}{\mathrm{x}^{2}}$
$\Rightarrow \frac{d^{2} y}{d x^{2}}=\frac{-2}{x^{2}}$
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