1. **State the problem:** We need to determine the derivative of the function $$f(x) = \frac{x^3 + 5}{x^2} - 25$$ with respect to $$x$$. 2. **Rewrite the function:** To differentiate easily, rewrite the function by dividing each term in the numerator by $$x^2$$: $$f(x) = \frac{x^3}{x^2} + \frac{5}{x^2} - 25 = x + 5x^{-2} - 25$$ 3. **Recall the derivative rules:** - The derivative of $$x^n$$ is $$nx^{n-1}$$. - The derivative of a constant is 0. 4. **Differentiate each term:** - Derivative of $$x$$ is $$1$$. - Derivative of $$5x^{-2}$$ is $$5 \times (-2) x^{-3} = -10x^{-3}$$. - Derivative of $$-25$$ is $$0$$. 5. **Combine the results:** $$f'(x) = 1 - 10x^{-3}$$ 6. **Rewrite the derivative in a simpler form:** $$f'(x) = 1 - \frac{10}{x^3}$$ **Final answer:** $$\boxed{f'(x) = 1 - \frac{10}{x^3}}$$