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The problem asks to find the derivative of the function \( y = (5x^2 + 3)(2x - 5) \) using the product rule. Let's start by recalling the product rule for derivatives.

### Product Rule:
If \( y = u(x) \cdot v(x) \), then:
\[
\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)
\]
where \( u(x) = 5x^2 + 3 \) and \( v(x) = 2x - 5 \).

### Step-by-step:

1. **First, differentiate \( u(x) = 5x^2 + 3 \):**
   \[
   u'(x) = \frac{d}{dx}(5x^2 + 3) = 10x
   \]
   
2. **Next, differentiate \( v(x) = 2x - 5 \):**
   \[
   v'(x) = \frac{d}{dx}(2x - 5) = 2
   \]

### Using the product rule:
\[
\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)
\]
Substitute the values:
\[
\frac{dy}{dx} = (10x)(2x - 5) + (5x^2 + 3)(2)
\]

Now let's simplify the terms:
- First term: \( (10x)(2x - 5) = 20x^2 - 50x \)
- Second term: \( (5x^2 + 3)(2) = 10x^2 + 6 \)

So the derivative becomes:
\[
\frac{dy}{dx} = 20x^2 - 50x + 10x^2 + 6 = 30x^2 - 50x + 6
\]

### Correct form of the product rule:
Looking at the answer choices, we need to match the way the product rule is applied. The correct answer is **Option D**, as it correctly uses the product rule format and represents the differentiation steps accurately:
\[
\frac{dy}{dx} = \left( \frac{d}{dx}(5x^2 + 3) \cdot (2x - 5) \right) + \left( \frac{d}{dx}(2x - 5) \cdot (5x^2 + 3) \right)
\]

Let me know if you'd like more details, or have any questions!

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Here are five related questions:
1. Can you apply the product rule to a different pair of functions?
2. What is the derivative of \( (3x^3 + 2)(4x - 1) \)?
3. What happens if you apply the quotient rule instead of the product rule to this function?
4. How would you find the second derivative of the given function \( y \)?
5. What is the chain rule and when would you use it in derivatives?

**Tip:** Always simplify your expressions after applying the product rule for a cleaner final result.

Use the product rule to find the derivative of the function. y = (5x^2 + 3)(2x - 5). What is the correct way of writing the derivative of y?

Learn how to find the derivative of y = (5x^2 + 3)(2x - 5) by applying the product rule. This calculus problem demonstrates key concepts like the product rule for derivatives, differentiation of polynomials, and simplification. Suitable for students in Grades 11-12 or early college calculus.

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