1. **Problem Statement:** Given the function $$f(x) = \frac{x^2 - 2}{x^4}$$, find its first derivative $$f'(x)$$ and second derivative $$f''(x)$$. 2. **Rewrite the function:** To differentiate easily, rewrite $$f(x)$$ as $$f(x) = (x^2 - 2) x^{-4} = x^{2-4} - 2x^{-4} = x^{-2} - 2x^{-4}$$. 3. **First derivative:** Use the power rule $$\frac{d}{dx} x^n = n x^{n-1}$$. $$f'(x) = \frac{d}{dx} (x^{-2}) - 2 \frac{d}{dx} (x^{-4}) = -2 x^{-3} - 2(-4) x^{-5} = -2 x^{-3} + 8 x^{-5}$$. 4. **Simplify first derivative:** $$f'(x) = -\frac{2}{x^3} + \frac{8}{x^5}$$. 5. **Second derivative:** Differentiate $$f'(x)$$ again: $$f''(x) = \frac{d}{dx} \left(-2 x^{-3} + 8 x^{-5} \right) = -2(-3) x^{-4} + 8(-5) x^{-6} = 6 x^{-4} - 40 x^{-6}$$. 6. **Simplify second derivative:** $$f''(x) = \frac{6}{x^4} - \frac{40}{x^6}$$. **Final answers:** $$f'(x) = -\frac{2}{x^3} + \frac{8}{x^5}$$ $$f''(x) = \frac{6}{x^4} - \frac{40}{x^6}$$