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We are given the function:

\[
y = \frac{8x^2 + 8x + 4}{\sqrt{x}} = \frac{8x^2 + 8x + 4}{x^{1/2}}
\]

We need to differentiate \(y\) with respect to \(x\). First, we will rewrite the function to make it easier to apply the derivative rules:

\[
y = (8x^2 + 8x + 4) \cdot x^{-1/2}
\]

Now, we can differentiate this using the product rule and the power rule. The product rule is:

\[
\frac{d}{dx}(uv) = u'v + uv'
\]

Where \(u = 8x^2 + 8x + 4\) and \(v = x^{-1/2}\). Let's compute the derivatives step by step.

1. **Differentiate \(u = 8x^2 + 8x + 4\):**

\[
u' = 16x + 8
\]

2. **Differentiate \(v = x^{-1/2}\):**

\[
v' = -\frac{1}{2} x^{-3/2}
\]

Now, apply the product rule:

\[
\frac{dy}{dx} = (16x + 8) \cdot x^{-1/2} + (8x^2 + 8x + 4) \cdot \left(-\frac{1}{2}x^{-3/2}\right)
\]

We can now simplify each term:

1. **First term:**

\[
(16x + 8) \cdot x^{-1/2} = 16x^{1 - 1/2} + 8x^{-1/2} = 16x^{1/2} + 8x^{-1/2}
\]

2. **Second term:**

\[
(8x^2 + 8x + 4) \cdot \left(-\frac{1}{2}x^{-3/2}\right) = -\frac{1}{2} \left(8x^2 + 8x + 4\right) \cdot x^{-3/2}
\]

Distribute \(x^{-3/2}\) to each term inside the parentheses:

\[
-\frac{1}{2} \left( 8x^2 \cdot x^{-3/2} + 8x \cdot x^{-3/2} + 4 \cdot x^{-3/2} \right)
\]

\[
= -\frac{1}{2} \left( 8x^{2 - 3/2} + 8x^{1 - 3/2} + 4x^{-3/2} \right)
\]

Simplifying the powers:

\[
= -\frac{1}{2} \left( 8x^{1/2} + 8x^{-1/2} + 4x^{-3/2} \right)
\]

\[
= -4x^{1/2} - 4x^{-1/2} - 2x^{-3/2}
\]

Now, combine the terms:

\[
\frac{dy}{dx} = (16x^{1/2} + 8x^{-1/2}) + (-4x^{1/2} - 4x^{-1/2} - 2x^{-3/2})
\]

\[
\frac{dy}{dx} = (16x^{1/2} - 4x^{1/2}) + (8x^{-1/2} - 4x^{-1/2}) - 2x^{-3/2}
\]

\[
\frac{dy}{dx} = 12x^{1/2} + 4x^{-1/2} - 2x^{-3/2}
\]

Thus, the derivative is:

\[
\boxed{\frac{dy}{dx} = 12x^{1/2} + 4x^{-1/2} - 2x^{-3/2}}
\]

Let me know if you'd like further details or have questions!

---

Here are 5 related questions to consider:
1. How do you apply the product rule in differentiation?
2. What are some common mistakes made when differentiating functions involving exponents?
3. How does the chain rule apply to functions with roots like \( \sqrt{x} \)?
4. How can you simplify expressions involving negative exponents?
5. How would you integrate a function like \(12x^{1/2} + 4x^{-1/2} - 2x^{-3/2}\)?

**Tip:** When differentiating products, remember to apply the product rule correctly and simplify the terms carefully!

Differentiate the function y = (8x^2 + 8x + 4) / √x.

Learn how to differentiate the function y = (8x^2 + 8x + 4) / √x using the product rule. This step-by-step solution involves the application of the product rule and power rule, simplifying complex terms. Suitable for students in Grades 11-12 studying advanced calculus.

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