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This set of exercises asks for the derivatives using the **product rule**. Here's the formula for the product rule:

\[
\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
\]

I will solve each one for you step by step:

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

- Let \( f(x) = 3x^2 + 2 \) and \( g(x) = 2x - 1 \).
- Now, compute \( f'(x) \) and \( g'(x) \):
  \[
  f'(x) = 6x \quad \text{and} \quad g'(x) = 2
  \]
- Using the product rule:
  \[
  \frac{dy}{dx} = (6x)(2x - 1) + (3x^2 + 2)(2)
  \]
  Simplify:
  \[
  = 12x^2 - 6x + 6x^2 + 4 = 18x^2 - 6x + 4
  \]

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

- Let \( f(x) = 5x^2 - 1 \) and \( g(x) = 4x + 3 \).
- Compute derivatives:
  \[
  f'(x) = 10x \quad \text{and} \quad g'(x) = 4
  \]
- Applying the product rule:
  \[
  \frac{dy}{dx} = (10x)(4x + 3) + (5x^2 - 1)(4)
  \]
  Simplify:
  \[
  = 40x^2 + 30x + 20x^2 - 4 = 60x^2 + 30x - 4
  \]

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

- First, write \( y \) as a product: \( y = (2x - 5)(2x - 5) \).
- Let \( f(x) = 2x - 5 \) and \( g(x) = 2x - 5 \).
- Compute derivatives:
  \[
  f'(x) = 2 \quad \text{and} \quad g'(x) = 2
  \]
- Apply the product rule:
  \[
  \frac{dy}{dx} = (2)(2x - 5) + (2x - 5)(2)
  \]
  Simplify:
  \[
  = 4(2x - 5) = 8x - 20
  \]

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

- Rewrite \( y = (7x - 6)(7x - 6) \).
- Let \( f(x) = 7x - 6 \) and \( g(x) = 7x - 6 \).
- Compute derivatives:
  \[
  f'(x) = 7 \quad \text{and} \quad g'(x) = 7
  \]
- Using the product rule:
  \[
  \frac{dy}{dx} = (7)(7x - 6) + (7x - 6)(7)
  \]
  Simplify:
  \[
  = 14(7x - 6) = 98x - 84
  \]

Would you like me to continue with the rest, or do you have any specific questions so far?

### Additional questions:
1. How does the product rule work when there are more than two functions?
2. Can the chain rule be applied in combination with the product rule?
3. What are the key differences between the product rule and quotient rule?
4. How do we approach higher-order derivatives using the product rule?
5. How would these problems change if one of the factors was a constant?

#### Tip: 
Always simplify the terms as much as possible after applying the product rule to avoid mistakes!

Use the product rule to find the derivative of the following functions.

Practice applying the product rule for differentiation with these exercises. This set includes derivatives of polynomials and more complex functions. Ideal for Grades 11-12 calculus students.

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