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!

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.