https://file.aiquickdraw.com/mathbot/file/1729962159090slwhz7q8.jpg

The equation given in the image is:

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

To find the derivative \( \frac{dy}{dx} \), let's apply the differentiation rules.

### Step-by-Step Solution:

1. **Differentiate \( y = 6 \) with respect to \( x \):**  
   Since \(6\) is a constant, its derivative is \(0\).

2. **Differentiate \(4x \cdot \csc(x)\):**  
   This part requires the product rule, where if \( u = 4x \) and \( v = \csc(x) \), then:
   \[
   \frac{d}{dx}(uv) = u'v + uv'
   \]

   - \( u = 4x \)  
     \( u' = 4 \)

   - \( v = \csc(x) \)  
     \( v' = -\csc(x) \cot(x) \) (using the derivative of \( \csc(x) \))

   Now, applying the product rule:
   \[
   \frac{d}{dx}(4x \cdot \csc(x)) = 4 \cdot \csc(x) + 4x \cdot (-\csc(x) \cot(x))
   \]

   Simplifying:
   \[
   = 4 \csc(x) - 4x \csc(x) \cot(x)
   \]

### Final Answer:

Thus, the derivative \( \frac{dy}{dx} \) is:
\[
\frac{dy}{dx} = 4 \csc(x) - 4x \csc(x) \cot(x)
\]

Would you like further details on any step, or have additional questions?

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Here are some related questions you might find interesting:
1. How do we apply the product rule to other trigonometric functions?
2. What are the common derivatives of trigonometric functions?
3. How does the chain rule affect derivatives involving trigonometric functions?
4. Can we apply implicit differentiation to trigonometric identities?
5. How does the cotangent function's derivative compare to other trig functions?

**Tip:** When differentiating products of functions, always apply the product rule carefully and simplify terms where possible.

Find the derivative of y = 6 + 4x * csc(x).

Learn how to differentiate y = 6 + 4x * csc(x) using the product rule and trigonometric differentiation techniques. This solution involves key calculus concepts and provides a detailed step-by-step approach suitable for students in Grades 11-12.

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