https://file.aiquickdraw.com/mathbot/file/1726255492351df0u6v1h.jpg

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!

Use the General Power Rule to find the derivative of f(x) = (6x - x^2)^3. The inside function is u = 6x - x^2.

Learn how to differentiate the function f(x) = (6x - x^2)^3 using the General Power Rule. This solution explains the application of the chain rule and includes the step-by-step process to compute the derivative. Suitable for high school calculus students.

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