Let's analyze the problem step by step.

We are given the function $f(x) = (6x - x^2)^3$ and asked to find its derivative $f'(x)$ using the General Power Rule (Chain Rule).

Step 1: Recognize the Outer and Inner Functions

• The outer function is $u^3$, where $u = 6x - x^2$.
• The inner function is $u = 6x - x^2$.

Step 2: Apply the Chain Rule

The Chain Rule states: $f'(x) = 3(6x - x^2)^2 \cdot \frac{d}{dx}(6x - x^2)$

Step 3: Differentiate the Inner Function

Now, we differentiate $6x - x^2$: $\frac{d}{dx}(6x - x^2) = 6 - 2x$

Step 4: Combine the Results

Thus, the derivative is: $f'(x) = 3(6x - x^2)^2 \cdot (6 - 2x)$

Completing the Template

In the boxes shown in the image:

• The first blank will have the constant $3$.
• The second blank will be the derivative of $6x - x^2$, which is $6 - 2x$.

So, the final expression becomes: $f'(x) = 3(6x - x^2)^2 \cdot (6 - 2x)$

Would you like further clarification on any part? Here are 5 additional related questions you might find useful:

1. How do you apply the chain rule to more complex functions?
2. What is the derivative of $(ax + b)^n$ in general?
3. Can you explain why the chain rule works?
4. How do you differentiate a product of two functions?
5. What is the difference between the chain rule and the product rule?

Tip: Always simplify the inner function before differentiating to avoid unnecessary complexity!