We are given the equation:

\[
(x - y)^3 - 9xy = 4
\]

To find \(\frac{dy}{dx}\), we will implicitly differentiate both sides of the equation with respect to \(x\).

### Step 1: Differentiate both sides with respect to \(x\)

The left side has two terms: \((x - y)^3\) and \(-9xy\).

#### Differentiating \((x - y)^3\):
Using the chain rule, we get:
\[
\frac{d}{dx}\left((x - y)^3\right) = 3(x - y)^2 \cdot \frac{d}{dx}(x - y)
\]
Now, differentiate \(x - y\) with respect to \(x\):
\[
\frac{d}{dx}(x - y) = 1 - \frac{dy}{dx}
\]
Thus,
\[
\frac{d}{dx}\left((x - y)^3\right) = 3(x - y)^2 \cdot (1 - \frac{dy}{dx})
\]

#### Differentiating \(-9xy\):
Using the product rule, we get:
\[
\frac{d}{dx}\left(-9xy\right) = -9\left(\frac{d}{dx}(x) \cdot y + x \cdot \frac{d}{dx}(y)\right)
\]
\[
\frac{d}{dx}(x) = 1 \quad \text{and} \quad \frac{d}{dx}(y) = \frac{dy}{dx}
\]
Thus,
\[
\frac{d}{dx}\left(-9xy\right) = -9(y + x\frac{dy}{dx})
\]

### Step 2: Differentiate the right side
Since 4 is a constant, its derivative is 0:
\[
\frac{d}{dx}(4) = 0
\]

### Step 3: Combine all terms
Now, we put everything together:
\[
3(x - y)^2(1 - \frac{dy}{dx}) - 9(y + x\frac{dy}{dx}) = 0
\]

### Step 4: Solve for \(\frac{dy}{dx}\)
Expand the equation:
\[
3(x - y)^2 - 3(x - y)^2 \frac{dy}{dx} - 9y - 9x \frac{dy}{dx} = 0
\]

Group the terms with \(\frac{dy}{dx}\):
\[
- 3(x - y)^2 \frac{dy}{dx} - 9x \frac{dy}{dx} = -3(x - y)^2 + 9y
\]

Factor out \(\frac{dy}{dx}\):
\[
\frac{dy}{dx}\left(-3(x - y)^2 - 9x\right) = -3(x - y)^2 + 9y
\]

Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{-3(x - y)^2 + 9y}{-3(x - y)^2 - 9x}
\]

### Final answer:
\[
\frac{dy}{dx} = \frac{9y - 3(x - y)^2}{-3(x - y)^2 - 9x}
\]

Would you like more details on any step, or do you have any questions?

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Here are 5 related questions you might find useful:
1. What is implicit differentiation, and when is it used?
2. How does the chain rule apply to composite functions?
3. Can you explain how the product rule works in differentiation?
4. How does differentiating with respect to \(x\) affect terms involving \(y\)?
5. What are some common mistakes to avoid in implicit differentiation?

**Tip**: Always simplify the equation as much as possible before differentiating—it can make the implicit differentiation easier to manage!

This page provides a step-by-step solution to the implicit differentiation problem (x - y)^3 - 9xy = 4. Learn how to apply the chain rule and product rule to find dy/dx, suitable for students in Grades 11-12 or those studying Calculus I.

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