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Let's solve this problem by applying the requested steps.

### Given:
\[
y = \left( 8x^2 + 3 \right) \left( 4x + 9 + \frac{5}{x} \right)
\]

#### (a) Apply the Product Rule to find \( y' \).

The Product Rule states:
\[
\frac{d}{dx} [u(x) \cdot v(x)] = u'(x)v(x) + u(x)v'(x)
\]
Here, let:
- \( u(x) = 8x^2 + 3 \)
- \( v(x) = 4x + 9 + \frac{5}{x} \)

Now, let's compute \( u'(x) \) and \( v'(x) \).

1. \( u'(x) = \frac{d}{dx}(8x^2 + 3) = 16x \)
2. \( v'(x) = \frac{d}{dx}\left( 4x + 9 + \frac{5}{x} \right) = 4 - \frac{5}{x^2} \)

Now, apply the Product Rule:
\[
y' = u'(x) v(x) + u(x) v'(x)
\]

Substitute the values:
\[
y' = (16x)\left( 4x + 9 + \frac{5}{x} \right) + \left( 8x^2 + 3 \right) \left( 4 - \frac{5}{x^2} \right)
\]

#### (b) Multiply the factors and simplify.

First, distribute \( 16x \) in the first term:
\[
16x \cdot (4x + 9 + \frac{5}{x}) = 16x \cdot 4x + 16x \cdot 9 + 16x \cdot \frac{5}{x}
\]
\[
= 64x^2 + 144x + 80
\]

Now, distribute \( (8x^2 + 3) \) in the second term:
\[
(8x^2 + 3) \cdot \left( 4 - \frac{5}{x^2} \right) = (8x^2)(4) + (8x^2)\left( -\frac{5}{x^2} \right) + (3)(4) + (3)\left( -\frac{5}{x^2} \right)
\]
\[
= 32x^2 - 40 + 12 - \frac{15}{x^2}
\]
\[
= 32x^2 - 28 - \frac{15}{x^2}
\]

Now combine everything:
\[
y' = (64x^2 + 144x + 80) + (32x^2 - 28 - \frac{15}{x^2})
\]
\[
y' = 64x^2 + 32x^2 + 144x + 80 - 28 - \frac{15}{x^2}
\]
\[
y' = 96x^2 + 144x + 52 - \frac{15}{x^2}
\]

So, the derivative \( y' \) is:
\[
y' = 96x^2 + 144x + 52 - \frac{15}{x^2}
\]

Would you like more details on any step?

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Here are five related questions to extend this knowledge:
1. What is the general form of the product rule for differentiation?
2. How would you differentiate more complex functions involving three or more terms using the product rule?
3. Can the quotient rule be used instead of the product rule in some cases, and if so, when?
4. How would the second derivative \( y'' \) be calculated for this function?
5. How can partial fractions be used to simplify similar expressions before differentiation?

**Tip**: Always simplify functions as much as possible before applying the product or quotient rules to make the differentiation easier.

Find y' by (a) applying the Product Rule and (b) multiplying the factors to produce a sum of simpler terms to differentiate. y = (8x^2 + 3)(4x + 9 + 5/x)

Learn how to find the derivative y' of the function y = (8x^2 + 3)(4x + 9 + 5/x) by applying the Product Rule and simplifying the expression. This problem involves using key differentiation techniques and algebraic simplification. Suitable for college-level calculus students.

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