To solve the problem of finding $f'(z)$ for the given function $f(z) = (5z^2 + 6z)(3z - 2)$, we need to follow these steps:

Step 1: Expand the Function

First, simplify the function by expanding the product: $f(z) = (5z^2 + 6z)(3z - 2)$ Apply the distributive property (FOIL method): $f(z) = 5z^2 \cdot 3z + 5z^2 \cdot (-2) + 6z \cdot 3z + 6z \cdot (-2)$ Simplify each term: $f(z) = 15z^3 - 10z^2 + 18z^2 - 12z$ Combine like terms: $f(z) = 15z^3 + 8z^2 - 12z$

Step 2: Find the Derivative $f'(z)$

Now differentiate each term with respect to $z$:

• The derivative of $15z^3$ is $45z^2$.
• The derivative of $8z^2$ is $16z$.
• The derivative of $-12z$ is $-12$.

So, the derivative $f'(z)$ is: $f'(z) = 45z^2 + 16z - 12$

Final Answer

The correct answer is $45z^2 + 16z - 12$.

Would you like more details or have any questions?

Here are some related questions to consider:

1. How do you multiply polynomials using the distributive property?
2. What is the FOIL method, and how is it used in polynomial multiplication?
3. How do you combine like terms in polynomial expressions?
4. What are the steps to differentiate a polynomial?
5. How would the derivative change if the original function included a higher power term, such as $z^4$?
6. What is the significance of the coefficient of $z^2$ in the derivative?
7. How does the power rule apply when differentiating polynomial terms?
8. Can you factor the original function before differentiating, and how would that affect the derivative?

Tip: Expanding a product before differentiating can simplify the process and reduce the likelihood of errors.