Let's work through the problem to find the correct choice.

We are asked to find the derivative of the function $h(z) = (z^3 - 2z^2 + 2z)(z - 6)$ in two different ways.

a) Using the Product Rule:

The Product Rule states that if we have a product of two functions $f(z) \cdot g(z)$, then: $\frac{d}{dz}[f(z) \cdot g(z)] = f'(z) \cdot g(z) + f(z) \cdot g'(z)$ Here, we can consider $f(z) = z^3 - 2z^2 + 2z$ and $g(z) = z - 6$.

1. First, find the derivative of $f(z)$: $f'(z) = \frac{d}{dz}[z^3 - 2z^2 + 2z] = 3z^2 - 4z + 2$

2. The derivative of $g(z) = z - 6$ is: $g'(z) = 1$

Now, apply the product rule: $h'(z) = f'(z) \cdot g(z) + f(z) \cdot g'(z)$ Substitute the values: $h'(z) = (3z^2 - 4z + 2)(z - 6) + (z^3 - 2z^2 + 2z)(1)$

This is the derivative using the product rule.

Answer Choices:

• The expression we derived matches Option D: $(z - 6)(3z^2 - 4z + 2) + (z^3 - 2z^2 + 2z)$

So, D is the correct option, and there are no further blanks to fill in. You can select this answer directly.

Would you like to see the solution for part (b) as well, or do you have further questions?

Follow-up Questions:

1. How does expanding the product help simplify the derivative?
2. What is the importance of the Product Rule in calculus?
3. Can you explain how the derivative of $f(z)$ was found?
4. How do you differentiate a product of functions generally?
5. How does this example show the symmetry of differentiation?

Tip: Always check the function you're differentiating for simplifications before using the product rule. It can sometimes save time!