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The exercises shown ask for the use of the **product rule** to differentiate the given functions. I will go through the first few exercises step by step to demonstrate how the product rule is applied.

### Recall the **Product Rule** for derivatives:
\[
\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)
\]

#### 1. \( y = (3x^2 + 2)(2x - 1) \)

Let \( f(x) = 3x^2 + 2 \) and \( g(x) = 2x - 1 \).
- \( f'(x) = 6x \)
- \( g'(x) = 2 \)

Using the product rule:
\[
y' = (6x)(2x - 1) + (3x^2 + 2)(2)
\]
Simplify each part:
\[
y' = 6x(2x - 1) + 2(3x^2 + 2)
\]
\[
y' = 12x^2 - 6x + 6x^2 + 4
\]
\[
y' = 18x^2 - 6x + 4
\]

#### 2. \( y = (5x^2 - 1)(4x + 3) \)

Let \( f(x) = 5x^2 - 1 \) and \( g(x) = 4x + 3 \).
- \( f'(x) = 10x \)
- \( g'(x) = 4 \)

Using the product rule:
\[
y' = (10x)(4x + 3) + (5x^2 - 1)(4)
\]
Simplify:
\[
y' = 10x(4x + 3) + 4(5x^2 - 1)
\]
\[
y' = 40x^2 + 30x + 20x^2 - 4
\]
\[
y' = 60x^2 + 30x - 4
\]

#### 3. \( y = (2x - 5)^2 \)

This is a special case because it is a single function squared. We can rewrite it as a product:
\[
y = (2x - 5)(2x - 5)
\]
Let \( f(x) = g(x) = 2x - 5 \).
- \( f'(x) = g'(x) = 2 \)

Using the product rule:
\[
y' = (2)(2x - 5) + (2x - 5)(2)
\]
Simplify:
\[
y' = 2(2x - 5) + 2(2x - 5)
\]
\[
y' = 4(2x - 5) = 8x - 20
\]

#### 4. \( y = (7x - 6)^2 \)

Again, this is a function squared. We can treat it as a product:
\[
y = (7x - 6)(7x - 6)
\]
Let \( f(x) = g(x) = 7x - 6 \).
- \( f'(x) = g'(x) = 7 \)

Using the product rule:
\[
y' = (7)(7x - 6) + (7x - 6)(7)
\]
Simplify:
\[
y' = 7(7x - 6) + 7(7x - 6)
\]
\[
y' = 14(7x - 6) = 98x - 84
\]

Would you like me to continue solving the rest of the exercises or provide more detailed explanations on the steps? 

### Follow-up questions to enhance understanding:
1. Can you apply the product rule to a more complex product like Exercise 9?
2. How does the product rule differ from the chain rule?
3. How would you approach differentiating higher-order products (e.g., Exercise 10)?
4. Could you try differentiating a sum of two products?
5. Can you explain why we need to simplify after applying the product rule?

### Tip:
Always simplify your final expression after using the product rule, as it can help you see patterns or common factors!

Use the product rule to find the derivative of the following. (Hint for Exercises 3–6: Write the quantity as a product.)

Explore detailed solutions using the product rule to differentiate various functions as shown in these calculus exercises. This guide is ideal for high school and college students looking to strengthen their understanding of derivatives.

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