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From the image, it looks like you are working on a problem related to derivatives. Specifically, you have a function \( g(x) = \frac{3x^2 - 6}{x^3 - 1} \), and you're being asked to find the derivative \( g'(x) \).

To solve this, we can apply the **quotient rule**, which is used for derivatives of fractions. The quotient rule is given by:

\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{v(x)^2}
\]

Where:
- \( u(x) = 3x^2 - 6 \)
- \( v(x) = x^3 - 1 \)

### Step 1: Compute the derivatives of the numerator and denominator
- \( u'(x) = \frac{d}{dx} (3x^2 - 6) = 6x \)
- \( v'(x) = \frac{d}{dx} (x^3 - 1) = 3x^2 \)

### Step 2: Apply the quotient rule
Using the quotient rule, we have:

\[
g'(x) = \frac{(x^3 - 1) \cdot 6x - (3x^2 - 6) \cdot 3x^2}{(x^3 - 1)^2}
\]

Now, we can expand and simplify the terms:

\[
g'(x) = \frac{6x(x^3 - 1) - 3x^2(3x^2 - 6)}{(x^3 - 1)^2}
\]

Simplifying the numerator:

\[
g'(x) = \frac{6x^4 - 6x - (9x^4 - 18x^2)}{(x^3 - 1)^2}
\]

\[
g'(x) = \frac{6x^4 - 6x - 9x^4 + 18x^2}{(x^3 - 1)^2}
\]

Combine like terms:

\[
g'(x) = \frac{-3x^4 + 18x^2 - 6x}{(x^3 - 1)^2}
\]

This is the derivative of the function \( g(x) \).

Would you like me to further explain or simplify any of these steps? Here are some related questions that could help deepen your understanding:

1. How does the quotient rule work for other types of functions?
2. Can you apply the chain rule to this type of derivative problem?
3. How does simplifying the final expression help in understanding the behavior of \( g'(x) \)?
4. What are some common mistakes when using the quotient rule in calculus?
5. How would you interpret the derivative \( g'(x) \) graphically?

**Tip**: When using the quotient rule, always remember to compute and simplify the derivatives of both the numerator and the denominator carefully before plugging them into the formula.

We consider the function g(x) = (3x^2 - 6) / (x^3 - 1). Find the derivative g'(x).

Learn how to find the derivative of the function g(x) = (3x^2 - 6) / (x^3 - 1) by applying the quotient rule. This step-by-step solution covers important calculus concepts, including the quotient rule, and is suitable for high school calculus students.

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