If y= _ex find d^2y dx^2 - Vidyadip

Question

If $\text{y}=\mid\log_\text{e}\text{x}\mid $ find $\frac{\text{d}^2\text{y}}{\text{dx}^2}$

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Answer

Here,
$\text{y}=\mid\log_\text{e}\text{x}\mid$
$=\begin{cases}-\log_\text{e}\text{x}&\text{if}&0<\text{x}< 1\\\log_\text{e}\text{x}&\text{if}&\text{x}>1\end{cases}$
Differentiating w.r.t.x, we get
$\frac{\text{dy}}{\text{dx}}=\begin{cases}\frac{-1}{\text{x}}&\text{if}&0<\text{x}<1\\\frac{1}{\text{x}}&\text{if}&\text{x}>1\end{cases}$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\begin{cases}\frac{1}{\text{x}^2}&\text{if}&0<\text{x}<1\\\frac{-1}{\text{x}^2}&\text{if}&\text{x}>1 \end{cases}$

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