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

Explanation

To solve the problem \( x^3 + y^3 = 3Axy \) and find \( \frac{dy}{dx} \), we will differentiate both sides of the equation with respect to \( x \).

### Step-by-Step Solution:

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

2. **Apply the Differentiation Rules**:
Using the power rule and the product rule:
- The derivative of \( x^3 \) is \( 3x^2 \).
- The derivative of \( y^3 \) is \( 3y^2 \frac{dy}{dx} \) (using the chain rule).
- For the right side, using the product rule on \( 3Axy \):
\[ \frac{d}{dx}(3Axy) = 3A \left( x \frac{dy}{dx} + y \right) \]
Putting it all together, we have:
\[ 3x^2 + 3y^2 \frac{dy}{dx} = 3A \left( x \frac{dy}{dx} + y \right) \]

3. **Rearranging the Equation**:
Now, we will rearrange the equation to isolate \( \frac{dy}{dx} \):
\[ 3y^2 \frac{dy}{dx} - 3Ax \frac{dy}{dx} = 3Ay - 3x^2 \]
Factor out \( \frac{dy}{dx} \):
\[ \left( 3y^2 - 3Ax \right) \frac{dy}{dx} = 3Ay - 3x^2 \]

4. **Solve for \( \frac{dy}{dx} \)**:
Now, divide both sides by \( 3y^2 - 3Ax \):
\[ \frac{dy}{dx} = \frac{3Ay - 3x^2}{3y^2 - 3Ax} \]
We can simplify this by factoring out 3:
\[ \frac{dy}{dx} = \frac{Ay - x^2}{y^2 - Ax} \]

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