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We are given the function \( f(x) = \frac{x^3}{6} - 2x \) and asked to find \( f'(-4) \), where \( f'(x) \) is the derivative of \( f(x) \).

### Step 1: Find the derivative \( f'(x) \)
To differentiate \( f(x) = \frac{x^3}{6} - 2x \), we apply the power rule and the constant rule:

- The derivative of \( \frac{x^3}{6} \) is \( \frac{3x^2}{6} = \frac{x^2}{2} \).
- The derivative of \( -2x \) is \( -2 \).

Thus, the derivative of \( f(x) \) is:
\[
f'(x) = \frac{x^2}{2} - 2
\]

### Step 2: Evaluate \( f'(-4) \)
Now, substitute \( x = -4 \) into \( f'(x) \):
\[
f'(-4) = \frac{(-4)^2}{2} - 2 = \frac{16}{2} - 2 = 8 - 2 = 6
\]

So, \( f'(-4) = 6 \).

### Final Answer:
\[
f'(-4) = 6
\]

Would you like further clarification on any steps, or have more questions?

Here are 5 related questions to deepen your understanding:
1. How do you apply the power rule in differentiation?
2. What are some common rules used in differentiation?
3. What happens if the given function has more terms or higher powers?
4. How can you interpret the derivative geometrically?
5. How would you find higher-order derivatives of this function?

**Tip**: Always double-check your algebraic simplifications when solving derivative problems to avoid mistakes.

This page provides a detailed solution for finding the derivative of the function f(x) = x^3/6 - 2x and evaluating f'(-4). Learn how to apply the power rule and constant rule to solve this calculus problem, ideal for students in Grades 11-12.

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