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

Answer

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

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

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

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

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

### Step 4: Factor out \(\frac{dy}{dx}\)
Factoring out \(\frac{dy}{dx}\) from the left side gives:
\[ \frac{dy}{dx}(3y^2 - 3ax) = 3ay - 3x^2 \]

### Step 5: Solve for \(\frac{dy}{dx}\)
Now, divide both sides by \(3y^2 - 3ax\) to solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{3ay - 3x^2}{3y^2 - 3ax} \]

### Step 6: Simplify the expression
We can simplify the expression by factoring out the common factor of \(3\):
\[ \frac{dy}{dx} = \frac{ay - x^2}{y^2 - ax} \]

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

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