To find the derivative of the function $h(z) = (z^3 - z^2 + 4z)(z - 6)$ using the Product Rule, we apply the following formula:

$\frac{d}{dz}[f(z)g(z)] = f'(z)g(z) + f(z)g'(z)$

Here:

• $f(z) = z^3 - z^2 + 4z$
• $g(z) = z - 6$

Step 1: Compute $f'(z)$ and $g'(z)$

• $f'(z) = 3z^2 - 2z + 4$
• $g'(z) = 1$ (since the derivative of $z$ is 1 and the derivative of $-6$ is 0)

Step 2: Apply the Product Rule

$h'(z) = f'(z)g(z) + f(z)g'(z)$

Substitute the expressions for $f(z)$, $g(z)$, $f'(z)$, and $g'(z)$:

$h'(z) = (3z^2 - 2z + 4)(z - 6) + (z^3 - z^2 + 4z)(1)$

Step 3: Simplify

Now, expand the terms: $(3z^2 - 2z + 4)(z - 6) = 3z^3 - 18z^2 - 2z^2 + 12z + 4z - 24 = 3z^3 - 20z^2 + 16z - 24$ $(z^3 - z^2 + 4z)(1) = z^3 - z^2 + 4z$

Now, sum the two results: $h'(z) = (3z^3 - 20z^2 + 16z - 24) + (z^3 - z^2 + 4z)$ $h'(z) = 4z^3 - 21z^2 + 20z - 24$

Conclusion:

The correct option is D, which matches the expanded form: $h'(z) = (z - 6)(3z^2 - 2z + 4) + (z^3 - z^2 + 4z)$

Do you need more details, or any further explanation?

Here are 5 related questions for practice:

1. What is the derivative of $h(z) = (z^4 - z^3)(z + 5)$ using the product rule?
2. How do you apply the product rule when there are more than two functions being multiplied?
3. What happens when one of the functions in a product rule is a constant?
4. Can you derive $h(z) = (e^z)(\sin z)$ using the product rule?
5. How can you verify a derivative by expanding the original product first?

Tip: Always simplify after applying the product rule to make it easier to combine terms.