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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.

Find the derivative of the function f(x) = (3x^2 - 2x^2 - 5x - 7) / x^2

Learn how to find the derivative of the rational function f(x) = (3x^2 - 2x^2 - 5x - 7) / x^2 with step-by-step simplification and application of the power rule. This calculus problem is ideal for students in Grades 11-12 or early college calculus.

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