If \(y=\displaystyle\frac{\log x}{x^2}\), then \(\displaystyle\frac{d^2y}{dx^2}\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{6\log x-5}{x^4}\)
By the quotient rule:
\( \displaystyle \frac{dy}{dx} = \frac{x^2\cdot\frac{1}{x} - \log x\cdot 2x}{x^4} \)
\( \displaystyle = \frac{x - 2x\log x}{x^4} \)
\( \displaystyle = \frac{1 - 2\log x}{x^3} \)
Applying the quotient rule again:
\( \displaystyle \frac{d^2y}{dx^2} = \frac{x^3\left(-\frac{2}{x}\right) - (1 - 2\log x)\cdot 3x^2}{x^6} \)
\( \displaystyle = \frac{-2x^2 - 3x^2 + 6x^2\log x}{x^6} \)
\( \displaystyle = \frac{6\log x - 5}{x^4} \)