Find the derivative of y w.r.t. x in each of the following :
`x^(2)=(x+3y)/(x-3y)`

Answer

To find the derivative of \( y \) with respect to \( x \) for the equation

\[ x^2 = \frac{x + 3y}{x - 3y} \]

we will follow these steps:

### Step 1: Cross-Multiply
First, we will cross-multiply to eliminate the fraction:

\[ x^2 (x - 3y) = x + 3y \]

### Step 2: Expand the Equation
Now, we will expand both sides:

\[ x^3 - 3x^2y = x + 3y \]

### Step 3: Rearrange the Equation
Next, we will rearrange the equation to group all terms involving \( y \) on one side:

\[ x^3 - x = 3y + 3x^2y \]

This can be rewritten as:

\[ x^3 - x = y(3 + 3x^2) \]

### Step 4: Differentiate Implicitly
Now, we will differentiate both sides with respect to \( x \):

\[ \frac{d}{dx}(x^3 - x) = \frac{d}{dx}(y(3 + 3x^2)) \]

Using the product rule on the right side:

\[ 3x^2 - 1 = (3 + 3x^2) \frac{dy}{dx} + y(6x) \]

### Step 5: Isolate \( \frac{dy}{dx} \)
Now, we will isolate \( \frac{dy}{dx} \):

\[ 3x^2 - 1 - 6xy = (3 + 3x^2) \frac{dy}{dx} \]

### Step 6: Solve for \( \frac{dy}{dx} \)
Now, we can solve for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{3x^2 - 1 - 6xy}{3 + 3x^2} \]

### Final Answer
Thus, the derivative of \( y \) with respect to \( x \) is:

\[ \frac{dy}{dx} = \frac{3x^2 - 1 - 6xy}{3 + 3x^2} \]

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