গণিত প্রশ্ন

AI-তৈরি নমুনা উত্তর

Given,
$y = \frac{\ln x}{x}$

Differentiating with respect to $x$ using the quotient rule:
$y_1 = \frac{d}{dx}\left(\frac{\ln x}{x}\right) = \frac{x \cdot \frac{d}{dx}(\ln x) - \ln x \cdot \frac{d}{dx}(x)}{x^2}$
$y_1 = \frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2} = \frac{1 - \ln x}{x^2}$

Differentiating again with respect to $x$:
$y_2 = \frac{d}{dx}(y_1) = \frac{x^2 \cdot \frac{d}{dx}(1 - \ln x) - (1 - \ln x) \cdot \frac{d}{dx}(x^2)}{(x^2)^2}$
$y_2 = \frac{x^2 \left(-\frac{1}{x}\right) - (1 - \ln x)(2x)}{x^4}$
$y_2 = \frac{-x - 2x + 2x\ln x}{x^4}$
$y_2 = \frac{-3x + 2x\ln x}{x^4} = \frac{x(2\ln x - 3)}{x^4}$
$y_2 = \frac{2\ln x - 3}{x^3}$

(Proved)