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The function given is:

\[
y = 6 + 4\sqrt{x} \cdot \csc(x)
\]

To find the derivative \( \frac{dy}{dx} \), we can proceed by differentiating each term:

1. The constant term \(6\) has a derivative of \(0\).
2. For the second term \(4\sqrt{x} \cdot \csc(x)\), we will use the product rule.

### Step-by-Step Solution

Given:
\[
y = 6 + 4\sqrt{x} \cdot \csc(x)
\]

The derivative \( \frac{dy}{dx} \) of \( y \) is:

\[
\frac{dy}{dx} = 0 + \frac{d}{dx} \left( 4\sqrt{x} \cdot \csc(x) \right)
\]

Using the product rule for \( u \cdot v \), where \( u = 4\sqrt{x} \) and \( v = \csc(x) \):

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

#### Step 1: Differentiate \( u = 4\sqrt{x} \)

1. Rewrite \( 4\sqrt{x} \) as \( 4x^{1/2} \).
2. The derivative of \( 4x^{1/2} \) with respect to \( x \) is:
   \[
   u' = 4 \cdot \frac{1}{2} x^{-\frac{1}{2}} = 2x^{-\frac{1}{2}} = \frac{2}{\sqrt{x}}
   \]

#### Step 2: Differentiate \( v = \csc(x) \)

The derivative of \( \csc(x) \) with respect to \( x \) is:
\[
v' = -\csc(x) \cot(x)
\]

#### Step 3: Apply the Product Rule

Now we can apply the product rule:
\[
\frac{dy}{dx} = \left( \frac{2}{\sqrt{x}} \right) \csc(x) + 4\sqrt{x} \left( -\csc(x) \cot(x) \right)
\]

Simplify each term:
\[
\frac{dy}{dx} = \frac{2 \csc(x)}{\sqrt{x}} - 4\sqrt{x} \csc(x) \cot(x)
\]

### Final Answer

\[
\frac{dy}{dx} = \frac{2 \csc(x)}{\sqrt{x}} - 4\sqrt{x} \csc(x) \cot(x)
\]

Would you like further details on any part of the calculation?

Here are some related questions for deeper understanding:

1. What is the product rule in calculus, and how does it apply to derivatives?
2. How can you simplify expressions involving trigonometric functions like \( \csc(x) \) and \( \cot(x) \)?
3. What are the practical applications of taking derivatives of functions involving trigonometric terms?
4. How does the chain rule relate to differentiating \( \sqrt{x} \)?
5. How do derivatives of trigonometric functions differ from other types of functions?

**Tip:** When differentiating complex functions, remember to identify each part carefully and consider which rules (like product or chain rule) apply.

Learn how to find the derivative of y = 6 + 4√x · csc(x) using the product rule and trigonometric differentiation. This detailed solution includes steps to differentiate the square root and cosecant functions, ideal for college calculus or advanced high school students.

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