`"If "x^(3)+y^(3)=3 axy," find "(dy)/(dx).`

Explanation

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

### Step 1: Differentiate both sides with respect to \( x \)

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 the differentiation rules

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

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

### Step 3: Rearrange the equation

Now we can simplify and rearrange the equation:
\[ 3x^2 + 3y^2 \frac{dy}{dx} = 3ay + 3ax \frac{dy}{dx} \]

Next, we group all the terms involving \( \frac{dy}{dx} \) on one side:
\[ 3y^2 \frac{dy}{dx} - 3ax \frac{dy}{dx} = 3ay - 3x^2 \]

### Step 4: Factor out \( \frac{dy}{dx} \)

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

### Step 5: Solve for \( \frac{dy}{dx} \)

Now we can isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{3ay - 3x^2}{3y^2 - 3ax} \]

Simplifying this, we have:
\[ \frac{dy}{dx} = \frac{ay - x^2}{y^2 - ax} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{ay - x^2}{y^2 - ax} \]

---