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The problem given is to find the derivative \(\frac{dy}{dx}\) if \(-y = \sin(x^3)\).

Let's solve this step by step.

### Step 1: Differentiate both sides with respect to \(x\).
The equation given is:
\[
-y = \sin(x^3)
\]

Differentiating both sides with respect to \(x\), we have:
\[
\frac{d(-y)}{dx} = \frac{d}{dx} \left(\sin(x^3)\right)
\]

### Step 2: Apply the chain rule.
- The derivative of \(-y\) with respect to \(x\) is \(-\frac{dy}{dx}\).
- For the right-hand side, we apply the chain rule:
  \[
  \frac{d}{dx} \left(\sin(x^3)\right) = \cos(x^3) \cdot \frac{d(x^3)}{dx}
  \]
  The derivative of \(x^3\) with respect to \(x\) is \(3x^2\).

So, we have:
\[
-\frac{dy}{dx} = \cos(x^3) \cdot 3x^2
\]

### Step 3: Solve for \(\frac{dy}{dx}\).
Multiply both sides by \(-1\):
\[
\frac{dy}{dx} = -3x^2 \cos(x^3)
\]

### Final Answer:
\[
\frac{dy}{dx} = -3x^2 \cos(x^3)
\]

Would you like any additional details on the steps, or do you have any other questions?

### Follow-up Questions:
1. What is the chain rule and how is it applied in differentiation?
2. How do you differentiate composite functions?
3. Can you explain how to differentiate trigonometric functions involving a composite argument like \(x^3\)?
4. What other methods can be used to find the derivative of more complex functions?
5. How does the derivative of a negative function, like \(-y\), affect the result?

### Tip:
When differentiating, always check if a function within the equation requires the use of the chain rule, especially when dealing with composite functions.

Learn how to differentiate the equation -y = sin(x^3) using the chain rule. This problem is suitable for students in Grades 11-12 and covers key concepts in differentiation and trigonometric functions.

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