To solve this problem, we need to find the derivative of the given function $y = (5x - 1)(2x + 3)$ using the Product Rule. The Product Rule states that if $y = uv$, where $u$ and $v$ are functions of $x$, then the derivative $y'$ is given by:

$y' = u'v + uv'$

Step 1: Identify $u$ and $v$

From the problem:

• $u = 5x - 1$
• $v = 2x + 3$

Step 2: Find the derivatives $u'$ and $v'$

• $u' = \frac{d}{dx}(5x - 1)$ $u' = 5$
• $v' = \frac{d}{dx}(2x + 3)$ $v' = 2$

Step 3: Apply the Product Rule

$y' = u'v + uv'$ Substitute $u$, $u'$, $v$, and $v'$ into the equation: $y' = (5)(2x + 3) + (5x - 1)(2)$

Step 4: Simplify each term

• First term: $(5)(2x + 3)$ $(5)(2x + 3) = 10x + 15$

• Second term: $(5x - 1)(2)$ $(5x - 1)(2) = 10x - 2$

Step 5: Combine like terms

$y' = 10x + 15 + 10x - 2 = 20x + 13$

Final Answer

The derivative $y'$ is $20x + 13$.

Looking at the options provided, this corresponds to the following choice:

• $\boxed{\text{Option B}}$

Would you like further details on this solution or have any questions?

Here are some related questions you might find helpful:

1. How do you apply the Product Rule to functions involving more complex expressions?
2. Why is it important to correctly identify $u$ and $v$ in the Product Rule?
3. How do you check your derivative results to ensure accuracy?
4. What is the next step after finding the derivative in a calculus problem?
5. How does the Product Rule relate to other derivative rules like the Quotient Rule?
6. Can you apply the Product Rule when one of the functions is a constant?
7. How would you find the second derivative of the function given in this problem?
8. What role do derivatives play in optimization problems?

Tip: When simplifying expressions after applying the Product Rule, be cautious of signs and coefficients to avoid small mistakes that can lead to incorrect final answers.