詳解：根據定義

\(f'(x) = \lim \limits_{\Delta x \to 0} \frac{{f(x + \Delta x) - f(x)}}{{\Delta x}}\)

\(f'(x) = \lim \limits_{\Delta x \to 0} \frac{{\frac{1}{{{{(x + \Delta x)}^2}}} - \frac{1}{{{x^2}}}}}{{\Delta x}} = \lim \limits_{\Delta x \to 0} \frac{{\frac{{ - 2x\Delta x - {{(\Delta x)}^2}}}{{{x^2}{{(x + \Delta x)}^2}}}}}{{\Delta x}} = \lim \limits_{\Delta x \to 0} \frac{{ - 2x - \Delta x}}{{{x^2}{{(x + \Delta x)}^2}}} = \frac{{ - 2x}}{{{x^2} \cdot {x^2}}} = \frac{{ - 2}}{{{x^3}}}\)