1. **State the problem:** Find the derivative of the function $$f(x) = x^5 - 3\sqrt{x} + \frac{1}{x^2}$$. 2. **Rewrite the function using exponents:** $$f(x) = x^5 - 3x^{\frac{1}{2}} + x^{-2}$$ 3. **Recall the power rule for derivatives:** If $$f(x) = x^n$$, then $$f'(x) = nx^{n-1}$$. 4. **Differentiate each term:** - For $$x^5$$, derivative is $$5x^{4}$$. - For $$-3x^{\frac{1}{2}}$$, derivative is $$-3 \times \frac{1}{2} x^{\frac{1}{2} - 1} = -\frac{3}{2} x^{-\frac{1}{2}}$$. - For $$x^{-2}$$, derivative is $$-2 x^{-3}$$. 5. **Combine the derivatives:** $$f'(x) = 5x^{4} - \frac{3}{2} x^{-\frac{1}{2}} - 2x^{-3}$$ 6. **Rewrite in radical and fraction form:** $$f'(x) = 5x^{4} - \frac{3}{2\sqrt{x}} - \frac{2}{x^{3}}$$ 7. **Check the options:** Option 1 matches exactly: $$f'(x) = 5x^{4} - \frac{3}{2\sqrt{x}} - \frac{2}{x^{3}}$$ **Final answer:** Option 1