https://file.aiquickdraw.com/mathbot/file/1727291377575615nu3tu.jpg

The given function is:

\[
g(s) = \frac{2s^2 - 9s + 6}{s^2}
\]

We need to find \( g'(s) \), the derivative of \( g(s) \).

### Step 1: Rewrite the function
First, simplify \( g(s) \) by splitting the terms:

\[
g(s) = \frac{2s^2}{s^2} - \frac{9s}{s^2} + \frac{6}{s^2}
\]
\[
g(s) = 2 - \frac{9}{s} + \frac{6}{s^2}
\]

### Step 2: Differentiate each term
Now, apply the power rule to differentiate each term.

1. \( \frac{d}{ds}(2) = 0 \)
2. \( \frac{d}{ds}\left(-\frac{9}{s}\right) = \frac{9}{s^2} \) (since \( \frac{d}{ds}\left(\frac{1}{s}\right) = -\frac{1}{s^2} \))
3. \( \frac{d}{ds}\left(\frac{6}{s^2}\right) = -\frac{12}{s^3} \) (using the power rule for \( s^{-2} \))

### Step 3: Combine the derivatives
Thus, the derivative is:

\[
g'(s) = 0 + \frac{9}{s^2} - \frac{12}{s^3}
\]
\[
g'(s) = \frac{9}{s^2} - \frac{12}{s^3}
\]

### Step 4: Check the answer choices
The correct answer is:

\[
\boxed{\frac{9}{s^2} - \frac{12}{s^3}}
\]

This matches the last option in the given multiple-choice list.

---

Do you need further details on any steps or have additional questions? Here are some related questions for practice:

1. How do you apply the power rule for negative exponents?
2. What happens when differentiating a constant term?
3. How does the quotient rule compare to simplifying first, as done here?
4. What is the derivative of \( \frac{1}{s^n} \) in general?
5. How do you find the second derivative of \( g(s) \)?

**Tip:** Always try to simplify expressions before differentiating to avoid complicated rules like the quotient rule.

If g(s) = (2s^2 - 9s + 6) / s^2, find g'(s).

Learn how to find the derivative of g(s) = (2s^2 - 9s + 6) / s^2 using calculus techniques like the power rule. This problem breaks down the simplification and differentiation steps and is suitable for advanced high school students in Grades 11-12.

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