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To find the derivative of the given function \( y \), we can apply the product rule and the chain rule from calculus.

The function given is:
\[ y = (3b + 2x) \sqrt{6x - x^2} \]

Let's denote:
\[ u = 3b + 2x \]
\[ v = \sqrt{6x - x^2} \]

The product rule states that:
\[ \frac{d}{dx}(uv) = u'v + uv' \]

First, we find the derivatives of \( u \) and \( v \).

1. **Derivative of \( u \):**
\[ u = 3b + 2x \]
Since \( b \) is a constant:
\[ u' = 2 \]

2. **Derivative of \( v \):**
\[ v = (6x - x^2)^{1/2} \]
Let \( w = 6x - x^2 \), then:
\[ v = w^{1/2} \]
Using the chain rule:
\[ \frac{d}{dx} (w^{1/2}) = \frac{1}{2} w^{-1/2} \cdot w' \]

Now, find \( w' \):
\[ w = 6x - x^2 \]
\[ w' = 6 - 2x \]

So,
\[ v' = \frac{1}{2} (6x - x^2)^{-1/2} \cdot (6 - 2x) \]
\[ v' = \frac{6 - 2x}{2\sqrt{6x - x^2}} \]

Now apply the product rule:
\[ y' = u'v + uv' \]
\[ y' = 2 \sqrt{6x - x^2} + (3b + 2x) \cdot \frac{6 - 2x}{2\sqrt{6x - x^2}} \]

Combine the terms:
\[ y' = 2 \sqrt{6x - x^2} + \frac{(3b + 2x)(6 - 2x)}{2\sqrt{6x - x^2}} \]

To simplify further, find a common denominator:
\[ y' = \frac{4(6x - x^2) + (3b + 2x)(6 - 2x)}{2\sqrt{6x - x^2}} \]

Let's simplify the numerator:
\[ 4(6x - x^2) = 24x - 4x^2 \]
\[ (3b + 2x)(6 - 2x) = 18b - 6bx + 12x - 4x^2 \]

Combining these:
\[ 24x - 4x^2 + 18b - 6bx + 12x - 4x^2 = 36x - 8x^2 + 18b - 6bx \]

So,
\[ y' = \frac{36x - 8x^2 + 18b - 6bx}{2\sqrt{6x - x^2}} \]

Thus, the simplified form of the derivative is:
\[ y' = \frac{36x - 8x^2 + 18b - 6bx}{2\sqrt{6x - x^2}} \]

Would you like more details or have any questions about this solution?

### Further Related Questions:
1. What are the steps to use the chain rule in differentiation?
2. How do you apply the product rule to functions involving square roots?
3. What are some common mistakes to avoid when differentiating complex functions?
4. Can you explain the physical interpretation of the derivative in this context?
5. How would the derivative change if \( b \) was a function of \( x \)?
6. What is the significance of finding the critical points of the derived function?
7. How do you simplify the final expression after applying the product and chain rules?
8. What are some advanced techniques for differentiating products of functions?

### Tip:
When differentiating complex expressions, breaking down the problem into smaller parts (using the chain rule and product rule) can simplify the process significantly.

Learn how to find the derivative and simplify the expression (3b + 2x) * sqrt(6x - x^2). This problem involves differentiation using the product rule and chain rule, focusing on algebraic manipulation and calculus techniques. Suitable for college-level students and advanced learners.

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