Find `dy/(dx)` if `x^(3)+y^(3)=3axy`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x^3 + y^3 = 3axy\), we will use implicit differentiation. Here’s the step-by-step solution:

### Step 1: Rewrite the equation
Start with the given equation:
\[ x^3 + y^3 - 3axy = 0 \]

### Step 2: Differentiate both sides with respect to \(x\)
Now, we differentiate both sides of the equation with respect to \(x\):
\[ \frac{d}{dx}(x^3) + \frac{d}{dx}(y^3) - \frac{d}{dx}(3axy) = 0 \]

### Step 3: Apply differentiation
Using the power rule and product rule, we differentiate each term:
- The derivative of \(x^3\) is \(3x^2\).
- For \(y^3\), we use the chain rule: \(\frac{d}{dx}(y^3) = 3y^2 \frac{dy}{dx}\).
- For \(3axy\), we apply the product rule:
\[ \frac{d}{dx}(3axy) = 3a \left( x \frac{dy}{dx} + y \right) \]

Putting it all together, we have:
\[ 3x^2 + 3y^2 \frac{dy}{dx} - 3a \left( x \frac{dy}{dx} + y \right) = 0 \]

### Step 4: Simplify the equation
Now, we can simplify the equation:
\[ 3x^2 + 3y^2 \frac{dy}{dx} - 3ax \frac{dy}{dx} - 3ay = 0 \]

### Step 5: Collect \(\frac{dy}{dx}\) terms
Rearranging gives:
\[ 3y^2 \frac{dy}{dx} - 3ax \frac{dy}{dx} = 3ay - 3x^2 \]

Factoring out \(\frac{dy}{dx}\):
\[ \frac{dy}{dx}(3y^2 - 3ax) = 3ay - 3x^2 \]

### Step 6: Solve for \(\frac{dy}{dx}\)
Now, divide both sides by \((3y^2 - 3ax)\):
\[ \frac{dy}{dx} = \frac{3ay - 3x^2}{3y^2 - 3ax} \]

### Step 7: Simplify the expression
We can simplify this further:
\[ \frac{dy}{dx} = \frac{ay - x^2}{y^2 - ax} \]

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