If y =[ ( x )]^2 then d^2 y d x^2 =

Question

If $y =[\log ( x )]^2$ then $\frac{d^2 y}{d x^2}=$

✓

Answer

$
\frac{2(1-\log x)}{x^2}
$
Hint:
$
\begin{aligned}
y & =(\log x)^2 \quad \therefore \frac{d y}{d x}=2 \log x \cdot \frac{d}{d x}(\log x) \\
& =2 \log x \times \frac{1}{x}=\frac{2 \log x}{x}
\end{aligned}
$
$
\text { and } \begin{aligned}
\frac{d^2 y}{d x^2} & =2 \frac{d}{d x}\left(\frac{\log x}{x}\right)
& =2\left(\frac{x \frac{d}{d x}(\log x)-(\log x) \cdot \frac{d}{d x}(x)}{x^2}\right) \\
& =2\left(\frac{x \times \frac{1}{x}-(\log x) \times 1}{x^2}\right) \\
& =\frac{2(1-\log x)}{x^2}
\end{aligned}
$

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