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To find the derivative of the function \( f(x) = (3x^2 - 7)(2x + 1) \), we can use the **product rule** for derivatives. The product rule states that for two functions \( u(x) \) and \( v(x) \), the derivative of their product is:

\[
(f \cdot g)' = f' \cdot g + f \cdot g'
\]

### Step-by-Step Solution

Let:
- \( u(x) = 3x^2 - 7 \)
- \( v(x) = 2x + 1 \)

#### Step 1: Differentiate \( u(x) \) and \( v(x) \) separately
1. \( u'(x) = \frac{d}{dx}(3x^2 - 7) = 6x \)
2. \( v'(x) = \frac{d}{dx}(2x + 1) = 2 \)

#### Step 2: Apply the product rule
Using \( f'(x) = u'(x)v(x) + u(x)v'(x) \), we get:

\[
f'(x) = (6x)(2x + 1) + (3x^2 - 7)(2)
\]

#### Step 3: Simplify each term
1. \( (6x)(2x + 1) = 12x^2 + 6x \)
2. \( (3x^2 - 7)(2) = 6x^2 - 14 \)

#### Step 4: Combine terms
\[
f'(x) = 12x^2 + 6x + 6x^2 - 14
\]
\[
f'(x) = 18x^2 + 6x - 14
\]

### Final Answer
\[
f'(x) = 18x^2 + 6x - 14
\]

Would you like more details or have any questions about the steps? 

Here are some related questions for further exploration:
1. How would you find the second derivative of this function?
2. Can we verify this derivative by expanding \( f(x) \) before differentiating?
3. How does the product rule apply if one of the factors was a constant?
4. What would happen if the function had three factors instead of two?
5. How does the chain rule differ from the product rule?

**Tip:** Always simplify each term in differentiation before combining for a clear answer.

Find the derivative of the function f(x) = (3x^2 - 7)(2x + 1). (Use symbolic notation and fractions where needed.)

Learn how to find the derivative of the function f(x) = (3x^2 - 7)(2x + 1) using the product rule. This solution provides a step-by-step approach, covering essential calculus concepts and rules. Suitable for Grade 12 and introductory college calculus students.

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