1. The problem asks us to find the derivative of the function $$f(x) = -\frac{1}{2x^3}$$ and express the answer using negative exponents. 2. First, rewrite the function using negative exponents: $$f(x) = -\frac{1}{2} x^{-3}$$ 3. Now, apply the power rule for derivatives, which states that $$\frac{d}{dx} x^n = n x^{n-1}$$. 4. Differentiate: $$f'(x) = -\frac{1}{2} \cdot (-3) x^{-3-1} = -\frac{1}{2} \cdot (-3) x^{-4}$$ 5. Simplify the constants: $$f'(x) = \frac{3}{2} x^{-4}$$ 6. Therefore, the derivative of the function is: $$f'(x) = \frac{3}{2} x^{-4}$$