If `(x^2+y^2)^2=x y` , find `(dy)/(dx)`

Explanation

To find \(\frac{dy}{dx}\) for the equation \((x^2 + y^2)^2 = xy\), we will follow these steps:

### Step 1: Expand the left side
We start with the equation:
\[ (x^2 + y^2)^2 = xy \]
Using the identity \((a + b)^2 = a^2 + b^2 + 2ab\), we expand the left side:
\[ x^4 + y^4 + 2x^2y^2 = xy \]

### Step 2: Differentiate both sides with respect to \(x\)
Now, we differentiate both sides of the equation with respect to \(x\):
\[ \frac{d}{dx}(x^4 + y^4 + 2x^2y^2) = \frac{d}{dx}(xy) \]

### Step 3: Apply the differentiation rules
Using the power rule and the product rule, we differentiate the left side:
- The derivative of \(x^4\) is \(4x^3\).
- The derivative of \(y^4\) using the chain rule is \(4y^3 \frac{dy}{dx}\).
- For \(2x^2y^2\), we apply the product rule:
\[ \frac{d}{dx}(2x^2y^2) = 2x^2 \cdot \frac{d}{dx}(y^2) + y^2 \cdot \frac{d}{dx}(2x^2) = 2x^2(2y \frac{dy}{dx}) + y^2(4x) = 4xy^2 + 4x^2y \frac{dy}{dx} \]

Putting it all together, we have:
\[ 4x^3 + 4y^3 \frac{dy}{dx} + 4xy^2 + 4x^2y \frac{dy}{dx} = y + x \frac{dy}{dx} \]

### Step 4: Rearranging the equation
Now, we rearrange the equation to isolate \(\frac{dy}{dx}\):
\[ 4y^3 \frac{dy}{dx} + 4x^2y \frac{dy}{dx} - x \frac{dy}{dx} = y - 4x^3 - 4xy^2 \]
Factoring out \(\frac{dy}{dx}\) from the left side:
\[ \left(4y^3 + 4x^2y - x\right) \frac{dy}{dx} = y - 4x^3 - 4xy^2 \]

### Step 5: Solve for \(\frac{dy}{dx}\)
Finally, we solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{y - 4x^3 - 4xy^2}{4y^3 + 4x^2y - x} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is given by:
\[ \frac{dy}{dx} = \frac{y - 4x^3 - 4xy^2}{4y^3 + 4x^2y - x} \]