We are given the equation:

\[
x^3 + \sin(xy) = 3
\]

We are tasked with finding \( \frac{dy}{dx} \), the derivative of \( y \) with respect to \( x \). To do this, we need to apply **implicit differentiation**.

### Step-by-step differentiation:

1. **Differentiate \( x^3 \)** with respect to \( x \):

\[
\frac{d}{dx}(x^3) = 3x^2
\]

2. **Differentiate \( \sin(xy) \)** using the **chain rule**. Since this is a product of \( x \) and \( y(x) \), we need to use the product rule and the chain rule:

\[
\frac{d}{dx}[\sin(xy)] = \cos(xy) \cdot \frac{d}{dx}(xy)
\]

Using the product rule on \( xy \):

\[
\frac{d}{dx}(xy) = x \frac{dy}{dx} + y
\]

So, we get:

\[
\frac{d}{dx}[\sin(xy)] = \cos(xy) \cdot (x \frac{dy}{dx} + y)
\]

3. **Differentiate the constant \( 3 \)**:

\[
\frac{d}{dx}(3) = 0
\]

Now, putting everything together:

\[
3x^2 + \cos(xy) \cdot (x \frac{dy}{dx} + y) = 0
\]

### Solve for \( \frac{dy}{dx} \):

First, move the \( 3x^2 \) to the other side:

\[
\cos(xy) \cdot (x \frac{dy}{dx} + y) = -3x^2
\]

Now, divide both sides by \( \cos(xy) \):

\[
x \frac{dy}{dx} + y = \frac{-3x^2}{\cos(xy)}
\]

Finally, isolate \( \frac{dy}{dx} \):

\[
x \frac{dy}{dx} = \frac{-3x^2}{\cos(xy)} - y
\]

\[
\frac{dy}{dx} = \frac{\frac{-3x^2}{\cos(xy)} - y}{x}
\]

This is the derivative of \( y \) with respect to \( x \).

### Final answer:

\[
\frac{dy}{dx} = \frac{-3x}{\cos(xy)} - \frac{y}{x}
\]

---

Would you like more details or have any specific questions?

### Related questions:
1. What is implicit differentiation and when is it used?
2. How do you apply the product rule to a function?
3. What is the chain rule and why is it important in differentiation?
4. Can you explain the meaning of the derivative \( \frac{dy}{dx} \) in this context?
5. How would the derivative change if we had \( \cos(xy) \) instead of \( \sin(xy) \)?

**Tip:** When solving problems with implicit differentiation, remember to apply the product and chain rules carefully, as variables can depend on each other.

Learn how to solve the implicit differentiation problem x³ + sin(xy) = 3 by applying the chain rule and product rule. This step-by-step solution includes key calculus concepts and is suitable for advanced high school or college-level students.

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