To find the first derivative of the equation \(x^2(x - y)^2 = x^2 - y^2\), we need to implicitly differentiate both sides with respect to \(x\). Since \(y\) is implicitly a function of \(x\), we’ll need to use the chain rule.

### Step 1: Differentiate both sides with respect to \(x\)
The equation is:
\[
x^2(x - y)^2 = x^2 - y^2
\]

#### Left-hand side (LHS):
Use the product rule to differentiate \(x^2(x - y)^2\):
\[
\frac{d}{dx}\left[x^2(x - y)^2\right] = \frac{d}{dx}\left(x^2\right)(x - y)^2 + x^2\frac{d}{dx}\left[(x - y)^2\right]
\]

- \(\frac{d}{dx}(x^2) = 2x\)
- Use the chain rule for \(\frac{d}{dx}\left[(x - y)^2\right]\):  
  \[
  \frac{d}{dx}\left[(x - y)^2\right] = 2(x - y)\left(\frac{d}{dx}(x - y)\right) = 2(x - y)(1 - \frac{dy}{dx})
  \]

So the derivative of the left-hand side is:
\[
2x(x - y)^2 + x^2 \cdot 2(x - y)\left(1 - \frac{dy}{dx}\right)
\]
Simplifying:
\[
2x(x - y)^2 + 2x^2(x - y)\left(1 - \frac{dy}{dx}\right)
\]

#### Right-hand side (RHS):
Now differentiate the right-hand side:
\[
\frac{d}{dx}(x^2 - y^2) = \frac{d}{dx}(x^2) - \frac{d}{dx}(y^2)
\]
- \(\frac{d}{dx}(x^2) = 2x\)
- \(\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}\) (by the chain rule)

So the derivative of the right-hand side is:
\[
2x - 2y \frac{dy}{dx}
\]

### Step 2: Set the derivatives equal to each other
Now, equate the derivatives from both sides:
\[
2x(x - y)^2 + 2x^2(x - y)\left(1 - \frac{dy}{dx}\right) = 2x - 2y \frac{dy}{dx}
\]

### Step 3: Solve for \(\frac{dy}{dx}\)
To solve for \(\frac{dy}{dx}\), collect all terms involving \(\frac{dy}{dx}\) on one side, and the rest on the other side.

#### Terms with \(\frac{dy}{dx}\) on the left-hand side:
\[
2x^2(x - y)(- \frac{dy}{dx}) = - 2y \frac{dy}{dx}
\]

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

Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{2x - 2x(x - y)^2}{-2x^2(x - y) + 2y}
\]
Simplify:
\[
\frac{dy}{dx} = \frac{x(1 - (x - y)^2)}{y - x^2(x - y)}
\]

### Final result:
\[
\boxed{\frac{dy}{dx} = \frac{x(1 - (x - y)^2)}{y - x^2(x - y)}}
\]

Would you like more details or have any specific questions about the process?

### Further Questions:
1. How do the product and chain rule work in differentiating composite functions?
2. What are some common techniques for solving implicit differentiation problems?
3. How do you check if your solution for \(\frac{dy}{dx}\) is correct?
4. What would happen if we tried to differentiate the equation as an explicit function \(y(x)\)?
5. Could this problem be simplified with a substitution?

### Tip:
When dealing with implicit differentiation, it's important to keep track of every term that involves \(y\) and apply the chain rule where necessary!

Find the first derivative of the equation x²(x - y)² = x² - y² using implicit differentiation. Learn step-by-step how to apply the product rule, chain rule, and solve for dy/dx. Suitable for advanced high school and college calculus students.

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