The problem asks for the derivative of the function:

$f(x) = \frac{3x^2 - 2x^2 - 5x - 7}{x^2}$

To solve this, let's first simplify the function. We can split the terms in the numerator by dividing each by $x^2$:

$f(x) = \frac{3x^2}{x^2} - \frac{2x^2}{x^2} - \frac{5x}{x^2} - \frac{7}{x^2}$

Simplifying each term:

$f(x) = 3 - 2 - \frac{5}{x} - \frac{7}{x^2}$

$f(x) = 1 - \frac{5}{x} - \frac{7}{x^2}$

Now, we can find the derivative $f'(x)$. Differentiate term by term:

$\frac{d}{dx}(1) = 0$

$\frac{d}{dx}\left(-\frac{5}{x}\right) = 5x^{-2}$

$\frac{d}{dx}\left(-\frac{7}{x^2}\right) = 14x^{-3}$

Thus, the derivative is:

$f'(x) = 5x^{-2} + 14x^{-3}$

Or, in more standard notation:

$f'(x) = \frac{5}{x^2} + \frac{14}{x^3}$

Now, let's check the provided responses:

• Response 1: $f'(x) = 6x - 3 + \frac{1}{x}$ → Incorrect.
• Response 2: $f'(x) = 6x - \frac{5}{x^2}$ → Incorrect.
• Response 3: $f'(x) = 6x + \frac{1}{x}$ → Incorrect.
• Response 4: $f'(x) = 6x - 3 + \frac{5}{x^2}$ → Incorrect.
• Response 5: $f'(x) = 6x - \frac{5}{x} + \frac{7}{x^2}$ → Incorrect.

None of the provided responses match the correct derivative.

Would you like further details or clarification?

Here are five follow-up questions:

1. What steps do you take when simplifying a rational function?
2. Can you explain how each term in the derivative was differentiated?
3. How would you handle a function with higher-degree terms?
4. Why do we rewrite terms like $\frac{5}{x}$ as $5x^{-1}$ before differentiating?
5. How would you check your derivative for correctness?

Tip: When simplifying complex functions, always check if terms can be reduced or factored out to make differentiation easier.