If `x^3+y^3=3a x y` , find `(dy)/(dx)`

Explanation

To find \(\frac{dy}{dx}\) for the equation \(x^3 + y^3 = 3axy\), we will differentiate both sides with respect to \(x\). Here are the steps:

### Step 1: Differentiate both sides
We start with the equation:
\[ x^3 + y^3 = 3axy \]
Now, we differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(x^3) + \frac{d}{dx}(y^3) = \frac{d}{dx}(3axy) \]

### Step 2: Apply differentiation rules
Using the power rule for \(x^3\) and the chain rule for \(y^3\), we get:
\[ 3x^2 + 3y^2 \frac{dy}{dx} = 3a \left( y + x \frac{dy}{dx} \right) \]

### Step 3: Expand the right-hand side
Expanding the right-hand side gives:
\[ 3x^2 + 3y^2 \frac{dy}{dx} = 3ay + 3ax \frac{dy}{dx} \]

### Step 4: Rearrange the equation
Now, we will rearrange the equation to isolate terms involving \(\frac{dy}{dx}\):
\[ 3y^2 \frac{dy}{dx} - 3ax \frac{dy}{dx} = 3ay - 3x^2 \]

### Step 5: Factor out \(\frac{dy}{dx}\)
Factoring \(\frac{dy}{dx}\) out from the left-hand side:
\[ \frac{dy}{dx} (3y^2 - 3ax) = 3ay - 3x^2 \]

### Step 6: Solve for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{3ay - 3x^2}{3y^2 - 3ax} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{ay - x^2}{y^2 - ax} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{ay - x^2}{y^2 - ax} \]