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Question
If \[y = \left| \log_e x \right|\] find\[\frac{d^2 y}{d x^2}\] ?
Solution
Here,
\[y = \left| \log_e x \right|\]
\[ =\begin{cases} - \log_e x & \text{ if } 0 < x < 1\ \\ \log_e x & \text { if }x > 1\end{cases}\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{d y}{d x} = \begin{cases}\frac{- 1}{x} & \text { if } 0 < x < 1\\ \frac{1}{x} & \text { if } x > 1\end{cases}\]
\[\text { Differentiating again w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = \begin{cases}\frac{1}{x^2} & \text { if } 0 < x < 1\\\frac{- 1}{x^2} & \text { if } x > 1\end{cases}\]
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