1. **State the problem:** Find the derivative of the function $$f(x) = -4x^6 + \frac{1}{x^7} - 5x^{\frac{5}{7}}$$. 2. **Recall the derivative rules:** - Power rule: $$\frac{d}{dx} x^n = nx^{n-1}$$ for any real number $$n$$. - Rewrite terms with negative exponents for easier differentiation: $$\frac{1}{x^7} = x^{-7}$$. 3. **Rewrite the function:** $$f(x) = -4x^6 + x^{-7} - 5x^{\frac{5}{7}}$$ 4. **Differentiate each term using the power rule:** - For $$-4x^6$$: $$\frac{d}{dx}(-4x^6) = -4 \times 6 x^{6-1} = -24x^5$$ - For $$x^{-7}$$: $$\frac{d}{dx} x^{-7} = -7 x^{-7-1} = -7x^{-8}$$ - For $$-5x^{\frac{5}{7}}$$: $$\frac{d}{dx} (-5x^{\frac{5}{7}}) = -5 \times \frac{5}{7} x^{\frac{5}{7} - 1} = -\frac{25}{7} x^{-\frac{2}{7}}$$ 5. **Combine the derivatives:** $$f'(x) = -24x^5 - 7x^{-8} - \frac{25}{7} x^{-\frac{2}{7}}$$ 6. **Rewrite negative exponents as fractions:** $$f'(x) = -24x^5 - \frac{7}{x^8} - \frac{25}{7} x^{-\frac{2}{7}}$$ 7. **Final answer:** $$f'(x) = -24x^5 - \frac{7}{x^8} - \frac{25}{7} x^{-\frac{2}{7}}$$ This matches the fourth option. **Explanation:** Each term was differentiated using the power rule, carefully handling negative and fractional exponents. Negative exponents correspond to reciprocal powers, and fractional exponents follow the same power rule.