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To find the derivative \(\frac{dy}{dx}\) for the given function \( y = \frac{\sin(3x)}{\sqrt{1 - x^3}} \), we will use the **quotient rule** along with the **chain rule**.

The quotient rule states that for a function \( y = \frac{u(x)}{v(x)} \), the derivative is:
\[
\frac{dy}{dx} = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}
\]
Where \( u(x) = \sin(3x) \) and \( v(x) = \sqrt{1 - x^3} \).

### Step 1: Differentiate the numerator \( u(x) = \sin(3x) \)
Using the chain rule:
\[
u'(x) = \cos(3x) \cdot \frac{d}{dx}(3x) = 3 \cos(3x)
\]

### Step 2: Differentiate the denominator \( v(x) = \sqrt{1 - x^3} \)
Using the chain rule again:
\[
v'(x) = \frac{1}{2\sqrt{1 - x^3}} \cdot \frac{d}{dx}(1 - x^3) = \frac{1}{2\sqrt{1 - x^3}} \cdot (-3x^2) = \frac{-3x^2}{2\sqrt{1 - x^3}}
\]

### Step 3: Apply the quotient rule
Now, applying the quotient rule:
\[
\frac{dy}{dx} = \frac{\sqrt{1 - x^3} \cdot 3\cos(3x) - \sin(3x) \cdot \left(\frac{-3x^2}{2\sqrt{1 - x^3}}\right)}{(1 - x^3)}
\]

### Step 4: Simplify the expression
The final expression for the derivative becomes:
\[
\frac{dy}{dx} = \frac{3 \cos(3x)\sqrt{1 - x^3} + \frac{3x^2 \sin(3x)}{2\sqrt{1 - x^3}}}{1 - x^3}
\]

This is the derivative of the given function.

Would you like further details on any of the steps? 

Here are 5 related questions to extend this topic:
1. How does the quotient rule apply when both the numerator and denominator are more complex functions?
2. Can you explain the chain rule's role in finding derivatives of composite functions like \(\sin(3x)\)?
3. How would this derivative change if the numerator were a different trigonometric function, like \(\cos(3x)\)?
4. What happens to the derivative when \( x = 0 \)? How can we compute it directly from the formula?
5. Can you solve this using implicit differentiation instead of the quotient rule?

**Tip:** When using the quotient rule, carefully differentiate both the numerator and denominator separately before combining them, as errors in one term can propagate through the entire solution.

Find the derivative dy/dx for y = sin(3x) / sqrt(1 - x^3).

Learn how to compute the derivative of y = sin(3x) / sqrt(1 - x^3) using the quotient and chain rule. The solution breaks down the process step-by-step, applying advanced calculus techniques for composite and rational functions. Suitable for undergraduate students.

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