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Let's find the derivative of the given function:

The function is:
\[
y = 3 \tan(x) + 3 \csc(x)
\]

To find the derivative \( \frac{dy}{dx} \), we will apply the sum rule, which states that the derivative of a sum is the sum of the derivatives.

### Step 1: Derivative of \( 3 \tan(x) \)
The derivative of \( \tan(x) \) is \( \sec^2(x) \). Thus:
\[
\frac{d}{dx} [3 \tan(x)] = 3 \sec^2(x)
\]

### Step 2: Derivative of \( 3 \csc(x) \)
The derivative of \( \csc(x) \) is \( -\csc(x) \cot(x) \). Thus:
\[
\frac{d}{dx} [3 \csc(x)] = 3 \left( -\csc(x) \cot(x) \right) = -3 \csc(x) \cot(x)
\]

### Step 3: Combine the results
Now, combine the derivatives from the previous steps:
\[
\frac{dy}{dx} = 3 \sec^2(x) - 3 \csc(x) \cot(x)
\]

Thus, the derivative is:
\[
\boxed{3 \sec^2(x) - 3 \csc(x) \cot(x)}
\]

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

Here are some follow-up questions to explore this further:
1. How does the chain rule apply when finding derivatives of composite functions?
2. What is the derivative of \( \sec(x) \) and \( \cot(x) \)?
3. Can you find the second derivative of the function \( y = 3 \tan(x) + 3 \csc(x) \)?
4. How would the derivative change if you had a constant added to the function \( y \)?
5. How can you graph the derivative of \( y = 3 \tan(x) + 3 \csc(x) \)?

**Tip:** Trigonometric derivatives often involve multiple identities. Remember to review key trigonometric functions and their derivatives for quicker calculations.

Find the derivative of the following function: y = 3 tan(x) + 3 csc(x)

This solution provides a step-by-step approach to finding the derivative of the trigonometric function y = 3 tan(x) + 3 csc(x). Key calculus concepts such as the sum rule and trigonometric derivatives are covered. Suitable for advanced high school or college-level students.

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