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To derive the given function \( y \) using the chain rule, let's first clarify the function:

\[ y = \frac{2}{(3 + \tan \theta)^3} \]

To find \(\frac{dy}{d\theta}\), we will apply the chain rule and other differentiation techniques. Let's proceed step-by-step.

### Step 1: Rewrite the function for easier differentiation
Rewrite the function in a form that is easier to differentiate:

\[ y = 2 (3 + \tan \theta)^{-3} \]

### Step 2: Apply the chain rule
We need to find the derivative of \( y \) with respect to \( \theta \):

\[ \frac{dy}{d\theta} = 2 \cdot \frac{d}{d\theta} \left( (3 + \tan \theta)^{-3} \right) \]

Let's focus on the inner function first, \((3 + \tan \theta)^{-3}\).

### Step 3: Differentiate the inner function using the chain rule
Let \( u = 3 + \tan \theta \). Then, the function becomes \( y = 2 u^{-3} \).

The derivative of \( y \) with respect to \( u \) is:

\[ \frac{dy}{du} = 2 \cdot (-3) u^{-4} = -6 u^{-4} \]

Now, we need to multiply this by the derivative of \( u \) with respect to \( \theta \).

### Step 4: Differentiate \( u \) with respect to \( \theta \)
Since \( u = 3 + \tan \theta \):

\[ \frac{du}{d\theta} = \sec^2 \theta \]

### Step 5: Combine the derivatives
Now, we combine these results using the chain rule:

\[ \frac{dy}{d\theta} = \frac{dy}{du} \cdot \frac{du}{d\theta} = -6 u^{-4} \cdot \sec^2 \theta \]

### Step 6: Substitute back \( u = 3 + \tan \theta \)
Finally, substitute \( u \) back into the expression:

\[ \frac{dy}{d\theta} = -6 (3 + \tan \theta)^{-4} \cdot \sec^2 \theta \]

So, the derivative of the given function \( y \) with respect to \( \theta \) is:

\[ \frac{dy}{d\theta} = -6 (3 + \tan \theta)^{-4} \cdot \sec^2 \theta \]

Would you like further details or have any questions on the steps involved?

Here are five questions you might consider asking next:
1. How do you differentiate trigonometric functions?
2. What is the chain rule and when is it used in differentiation?
3. How do you simplify expressions after differentiation?
4. Can you provide examples of more complex chain rule applications?
5. How does the derivative change if the function inside the chain rule is different?

**Tip:** Always rewrite complicated functions in simpler forms before applying differentiation rules.

Learn how to find the derivative of a function using the chain rule with this detailed step-by-step guide. This problem involves differentiating a complex trigonometric function, emphasizing the application of the chain rule in calculus. Suitable for advanced high school students and calculus enthusiasts.

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