Question

If $y=\frac{\log x}{x}$, show that $\frac{d^{2}y}{d{x}^2}=\frac{2\log x-3}{x^3}$.

Answer 1

Here,

$$\begin{gathered} y=\frac{\log x}{x} \\ \mathrm{Differentiating}\mathrm{w}.\mathrm{r}.\mathrm{t}.\mathit{}x,\mathrm{we}\mathrm{get} \\ \frac{\mathit{d}y}{\mathit{d}x}=\frac{1-\log x}{x^2} \\ \mathrm{Differentiating}\mathrm{again}\mathrm{w}.\mathrm{r}.\mathrm{t}.\mathit{}x,\mathrm{we}\mathrm{get} \\ \frac{{\mathit{d}}^{2}y}{\mathit{d}{x}^2}=\frac{-x-2x\left(1-\log x\right)}{x^4} \\ =\frac{-x-2x+2x\log x}{x^4} \\ =\frac{-3+2\log x}{x^3} \\ =\frac{2\log x-3}{x^3} \end{gathered}$$

Hence proved.