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.