`(x^(2)+y^(2))^(2) =xy`

Explanation

To solve the equation \((x^2 + y^2)^2 = xy\) and find \(\frac{dy}{dx}\), we will differentiate both sides with respect to \(x\).

### Step-by-step Solution:

1. **Differentiate both sides**:
We start with the equation:
\[ (x^2 + y^2)^2 = xy \]
We will differentiate both sides with respect to \(x\).

2. **Apply the chain rule on the left side**:
Using the chain rule, the left side becomes:
\[ \frac{d}{dx}[(x^2 + y^2)^2] = 2(x^2 + y^2) \cdot \frac{d}{dx}(x^2 + y^2) \]
Now, differentiate \(x^2 + y^2\):
\[ \frac{d}{dx}(x^2 + y^2) = 2x + 2y \frac{dy}{dx} \]
Thus, the left side becomes:
\[ 2(x^2 + y^2)(2x + 2y \frac{dy}{dx}) \]

3. **Differentiate the right side**:
For the right side \(xy\), we apply the product rule:
\[ \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \]

4. **Set the derivatives equal**:
Now we set the derivatives from both sides equal to each other:
\[ 2(x^2 + y^2)(2x + 2y \frac{dy}{dx}) = x \frac{dy}{dx} + y \]

5. **Expand and rearrange the equation**:
Expanding the left side gives:
\[ 4(x^2 + y^2)(x + y \frac{dy}{dx}) = x \frac{dy}{dx} + y \]
Rearranging terms, we get:
\[ 4(x^2 + y^2)xy \frac{dy}{dx} - x \frac{dy}{dx} = y - 4(x^2 + y^2)x \]

6. **Factor out \(\frac{dy}{dx}\)**:
Factoring out \(\frac{dy}{dx}\) from the left side:
\[ \left(4(x^2 + y^2)xy - x\right) \frac{dy}{dx} = y - 4(x^2 + y^2)x \]

7. **Solve for \(\frac{dy}{dx}\)**:
Finally, we isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{y - 4(x^2 + y^2)x}{4(x^2 + y^2)xy - x} \]

### Final Result:
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{y - 4(x^2 + y^2)x}{4(x^2 + y^2)xy - x} \]