Question

If $y=\left|{\log }_{e}x\right|$, find $\frac{d^{2}y}{d{x}^2}$.

Answer 1

Here,

$$\begin{gathered} y=\left|{\log }_{e}x\right| \\ =\left\{\begin{array}{l}-{\log }_{e}x\mathrm{if}0<x<1 \\ \\ {\log }_{e}x\mathrm{if}x>1\end{array}\right. \\ \mathrm{Differentiating}\mathrm{w}.\mathrm{r}.\mathrm{t}.x\mathit{,}\mathrm{we}\mathrm{get} \\ \frac{\mathit{d}y}{\mathit{d}x}=\left\{\begin{array}{l}\frac{-1}{x}\mathrm{if}0<x<1\\ \frac{1}{x}\mathrm{if}x>1\end{array}\right. \\ \mathrm{Differentiating}\mathrm{again}\mathrm{w}.\mathrm{r}.\mathrm{t}.x\mathit{,}\mathrm{we}\mathrm{get} \\ \frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^{\mathit{2}}}=\left\{\begin{array}{l}\frac{1}{x^2}\mathrm{if}0<x<1\\ \frac{-1}{x^2}\mathrm{if}x>1\end{array}\right. \end{gathered}$$