If \( y=\displaystyle\frac{log_e x}{x} \), then \( \displaystyle\frac{d^2y}{dx^2} \) is :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{2\,log_e\,x-3}{x^3} \)
By the quotient rule,
\( \displaystyle \frac{dy}{dx}=\frac{x\cdot\frac{1}{x}-\log_e x\cdot 1}{x^2} \)
\( \displaystyle =\frac{1-\log_e x}{x^2} \)
Applying the quotient rule again,
\( \displaystyle \frac{d^2y}{dx^2}=\frac{x^2\left(-\frac{1}{x}\right)-(1-\log_e x)\cdot 2x}{x^4} \)
\( \displaystyle =\frac{-x-2x+2x\log_e x}{x^4} \)
\( \displaystyle =\frac{2\log_e x-3}{x^3} \)