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,

\[y = \frac{\log x}{x}\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{d y}{d x} = \frac{1 - \log x}{x^2}\]
\[\text { Differentiating again w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = \frac{- x - 2x\left( 1 - \log x \right)}{x^4}\]
\[ = \frac{- x - 2x + 2x\log x}{x^4}\]
\[ = \frac{- 3 + 2\log x}{x^3}\]
\[ = \frac{2\log x - 3}{x^3}\]

Hence proved.