1. **Problem statement:** Find the derivative of the function $$f(x) = \frac{\ln x}{x^2}$$. 2. **Formula used:** We will use the quotient rule for derivatives, which states: $$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$$ where $u = \ln x$ and $v = x^2$. 3. **Calculate derivatives of numerator and denominator:** - Derivative of $u = \ln x$ is $u' = \frac{1}{x}$. - Derivative of $v = x^2$ is $v' = 2x$. 4. **Apply the quotient rule:** $$f'(x) = \frac{\frac{1}{x} \cdot x^2 - \ln x \cdot 2x}{(x^2)^2}$$ 5. **Simplify numerator:** $$\frac{1}{x} \cdot x^2 = x$$ So numerator becomes: $$x - 2x \ln x = x(1 - 2 \ln x)$$ 6. **Simplify denominator:** $$(x^2)^2 = x^4$$ 7. **Final derivative:** $$f'(x) = \frac{x(1 - 2 \ln x)}{x^4} = \frac{1 - 2 \ln x}{x^3}$$ **Answer:** $$\boxed{f'(x) = \frac{1 - 2 \ln x}{x^3}}$$