1. **State the problem:** Find the derivative $f'(x)$ of the function $f(x) = -4x^8$. 2. **Recall the power rule for derivatives:** If $f(x) = ax^n$, then $f'(x) = a n x^{n-1}$. 3. **Apply the power rule:** Here, $a = -4$ and $n = 8$, so $$f'(x) = -4 \times 8 \times x^{8-1} = -32x^7.$$ 4. **Final answer:** $$f'(x) = -32x^7.$$