To find the derivative of \( \frac{\sqrt{3 - x^2}}{3 + x} \), we'll need to apply both the **quotient rule** and the **chain rule**.

### Steps:

1. Let \( f(x) = \sqrt{3 - x^2} \) and \( g(x) = 3 + x \).
2. Use the quotient rule, which states:

\[
\left( \frac{f(x)}{g(x)} \right)' = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}
\]

3. Find \( f'(x) \) using the chain rule for \( f(x) = (3 - x^2)^{1/2} \):

\[
f'(x) = \frac{1}{2}(3 - x^2)^{-1/2} \cdot (-2x) = \frac{-x}{\sqrt{3 - x^2}}
\]

4. Find \( g'(x) \):

\[
g'(x) = 1
\]

5. Apply the quotient rule:

\[
\left( \frac{\sqrt{3 - x^2}}{3 + x} \right)' = \frac{(3 + x) \cdot \frac{-x}{\sqrt{3 - x^2}} - \sqrt{3 - x^2} \cdot 1}{(3 + x)^2}
\]

6. Simplify the expression:

\[
= \frac{(3 + x) \cdot \frac{-x}{\sqrt{3 - x^2}} - \sqrt{3 - x^2}}{(3 + x)^2}
\]

7. Combine terms:

\[
= \frac{-x(3 + x) - (3 - x^2)}{(3 + x)^2 \sqrt{3 - x^2}}
\]

\[
= \frac{-3x - x^2 - 3 + x^2}{(3 + x)^2 \sqrt{3 - x^2}}
\]

\[
= \frac{-3(x + 1)}{(3 + x)^2 \sqrt{3 - x^2}}
\]

Thus, the derivative is:

\[
\frac{d}{dx} \left( \frac{\sqrt{3 - x^2}}{3 + x} \right) = \frac{-3(x + 1)}{(3 + x)^2 \sqrt{3 - x^2}}
\]

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Do you want further details or have any questions?

Here are five related questions to expand your understanding:
1. How do you apply the product rule and chain rule together in derivatives?
2. What is the geometrical interpretation of the derivative of a function?
3. How does the quotient rule compare to the product rule?
4. Can you find the second derivative of the same function?
5. How would this derivative behave as \(x\) approaches 3?

**Tip:** Always simplify expressions before applying derivative rules to make the process easier.

Explore the step-by-step process of differentiating (sqrt(3 - x^2))/(3 + x) using the quotient and chain rules. This calculus problem involves applying key derivative rules to simplify the expression, suitable for undergraduate students in Calculus I or II.

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