To find the derivative of the given function:

$f(x) = \left(3x^3 - 4x - 8\right)^2$

we will need to use the chain rule. Here's the step-by-step process:

Step 1: Identify the outer and inner functions.

• Outer function: $(u(x))^2$, where $u(x) = 3x^3 - 4x - 8$
• Inner function: $u(x) = 3x^3 - 4x - 8$

Step 2: Apply the chain rule.

The chain rule states:

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

In our case, this becomes:

$f'(x) = 2(3x^3 - 4x - 8) \cdot \frac{d}{dx}(3x^3 - 4x - 8)$

Step 3: Differentiate the inner function.

The derivative of the inner function $3x^3 - 4x - 8$ is:

$\frac{d}{dx}(3x^3 - 4x - 8) = 9x^2 - 4$

Step 4: Combine everything.

Now, substitute the derivative of the inner function back into the chain rule expression:

$f'(x) = 2(3x^3 - 4x - 8) \cdot (9x^2 - 4)$

This is the derivative of the function.

Would you like to see further simplification or have more details?

Here are some related questions:

1. Can you simplify the derivative further?
2. How is the chain rule used in differentiation?
3. What happens if the function inside the square is different?
4. Can you apply the product rule here instead?
5. How do you apply the chain rule to trigonometric functions?

Tip: When applying the chain rule, make sure to differentiate both the outer and inner functions properly before combining them.