Question: Find the derivative of the function.
f(x) = \frac{x^3 - 9x^2 + 7}{x^2}
1. **State the problem:** Find the derivative of the function $$f(x) = \frac{x^3 - 9x^2 + 7}{x^2}$$.
2. **Rewrite the function:** Simplify the function by dividing each term in the numerator by $$x^2$$:
$$f(x) = \frac{x^3}{x^2} - \frac{9x^2}{x^2} + \frac{7}{x^2} = x - 9 + 7x^{-2}$$
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:**
- $$\frac{d}{dx}(x) = 1$$
- $$\frac{d}{dx}(-9) = 0$$
- $$\frac{d}{dx}(7x^{-2}) = 7 \cdot (-2)x^{-3} = -14x^{-3}$$
5. **Combine the derivatives:**
$$f'(x) = 1 + 0 - 14x^{-3} = 1 - 14x^{-3}$$
6. **Rewrite the derivative in fraction form:**
$$f'(x) = 1 - \frac{14}{x^3}$$
**Final answer:**
$$\boxed{f'(x) = 1 - \frac{14}{x^3}}$$