Let's solve the equation \( x^2 = \frac{x - y}{x + y} \) by finding \(\frac{dy}{dx}\) using implicit differentiation.

### Step 1: Rewrite the equation
We have:
\[
x^2 = \frac{x - y}{x + y}
\]

### Step 2: Differentiate both sides with respect to \(x\)
We apply the differentiation rules to both sides. 

On the left side, differentiate \(x^2\) with respect to \(x\):
\[
\frac{d}{dx}(x^2) = 2x
\]

On the right side, we use the quotient rule for differentiation:
\[
\frac{d}{dx}\left(\frac{x - y}{x + y}\right) = \frac{(x + y)\frac{d}{dx}(x - y) - (x - y)\frac{d}{dx}(x + y)}{(x + y)^2}
\]
Now, differentiate \(x - y\) and \(x + y\) with respect to \(x\):
\[
\frac{d}{dx}(x - y) = 1 - \frac{dy}{dx}, \quad \frac{d}{dx}(x + y) = 1 + \frac{dy}{dx}
\]
Substitute these into the quotient rule expression:
\[
\frac{d}{dx}\left(\frac{x - y}{x + y}\right) = \frac{(x + y)(1 - \frac{dy}{dx}) - (x - y)(1 + \frac{dy}{dx})}{(x + y)^2}
\]

### Step 3: Simplify the expression
Expand and simplify the numerator:
\[
(x + y)(1 - \frac{dy}{dx}) = x + y - (x + y)\frac{dy}{dx}
\]
\[
(x - y)(1 + \frac{dy}{dx}) = x - y + (x - y)\frac{dy}{dx}
\]
Subtract these two:
\[
(x + y - (x + y)\frac{dy}{dx}) - (x - y + (x - y)\frac{dy}{dx})
\]
Simplify further:
\[
x + y - x + y - (x + y)\frac{dy}{dx} - (x - y)\frac{dy}{dx} = 2y - 2x\frac{dy}{dx}
\]
Thus, the derivative on the right-hand side becomes:
\[
\frac{d}{dx}\left(\frac{x - y}{x + y}\right) = \frac{2y - 2x\frac{dy}{dx}}{(x + y)^2}
\]

### Step 4: Set the derivatives equal
Now, equate the derivative of the left side with the derivative of the right side:
\[
2x = \frac{2y - 2x\frac{dy}{dx}}{(x + y)^2}
\]

### Step 5: Solve for \(\frac{dy}{dx}\)
Multiply both sides by \((x + y)^2\) to eliminate the fraction:
\[
2x(x + y)^2 = 2y - 2x\frac{dy}{dx}
\]
Divide by 2:
\[
x(x + y)^2 = y - x\frac{dy}{dx}
\]
Rearrange to solve for \(\frac{dy}{dx}\):
\[
x\frac{dy}{dx} = y - x(x + y)^2
\]
\[
\frac{dy}{dx} = \frac{y - x(x + y)^2}{x}
\]

So, the derivative is:
\[
\frac{dy}{dx} = \frac{y - x(x + y)^2}{x}
\]

Would you like more details or have any questions?

Here are 8 related problems you can try:
1. Differentiate \( \frac{x + y}{x - y} = x^2 \) implicitly.
2. Solve \( x^3 + y^3 = 3xy \) using implicit differentiation.
3. Differentiate \( y = \frac{1}{x + y} \).
4. Find \(\frac{dy}{dx}\) for \( e^{xy} = x + y \).
5. Solve \( x^2 + y^2 + 2xy = 7 \) implicitly.
6. Differentiate \( x\sin(y) + y\cos(x) = 1 \) implicitly.
7. Find \(\frac{dy}{dx}\) for \( x^3 + y^3 + 3xy = 0 \).
8. Solve \( \ln(x + y) = xy \) implicitly.

**Tip:** Be careful when applying the quotient rule; simplify carefully to avoid mistakes in the differentiation process.

Learn how to solve the equation x^2 = (x - y)/(x + y) using implicit differentiation. This problem covers advanced concepts in calculus, including implicit differentiation and the quotient rule. Suitable for advanced high school students or college-level calculus students.

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