To find \(\frac{dy}{dx}\) from the equation \( x^3 + \sin(xy) = 3 \), we will use **implicit differentiation**. 

### Steps:

1. **Differentiate both sides of the equation with respect to \(x\):**

\[
\frac{d}{dx}(x^3 + \sin(xy)) = \frac{d}{dx}(3)
\]

The right-hand side, being a constant (3), has a derivative of 0:

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

Now, let's differentiate the left-hand side.

2. **Differentiate \( x^3 \):**

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

3. **Differentiate \( \sin(xy) \):**

We need to use the **chain rule** and **product rule** here. Consider \( \sin(xy) \) as a composite function. First, differentiate the outer function (sin), then the inner function \(xy\):

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

Now, apply the **product rule** to \( xy \):

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

Thus, the derivative of \( \sin(xy) \) is:

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

4. **Putting it all together:**

Now, we have:

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

5. **Solve for \(\frac{dy}{dx}\):**

Isolate the term involving \(\frac{dy}{dx}\):

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

Now, expand the left-hand side:

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

Next, isolate the term with \(\frac{dy}{dx}\):

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

Finally, solve for \(\frac{dy}{dx}\):

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

### Final answer:

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

Would you like further clarification or details?

### Related Questions:
1. What is implicit differentiation, and why do we use it in this case?
2. How does the product rule apply when differentiating \(xy\)?
3. Can we solve for \(y\) explicitly from \( x^3 + \sin(xy) = 3 \)?
4. How does the chain rule work when differentiating composite functions like \( \sin(xy) \)?
5. What would change if the equation were \( x^3 + \cos(xy) = 3 \)?

**Tip:** Implicit differentiation is useful when dealing with equations where \(y\) is a function of \(x\) but is not easily isolated.

Learn how to implicitly differentiate the equation x³ + sin(xy) = 3 to find dy/dx. This problem demonstrates key calculus techniques like implicit differentiation, the chain rule, and the product rule. Suitable for advanced high school students (Grades 11-12).

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