Find the derivative of y w.r.t. x in each of the following :
`x^(2)+y^(2)-3xy+1`

Answer

To find the derivative of \( y \) with respect to \( x \) for the equation \( x^2 + y^2 - 3xy + 1 = 0 \), we will use implicit differentiation. Here are the steps to solve the problem:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ x^2 + y^2 - 3xy + 1 = 0 \]
Now we differentiate each term with respect to \( x \).

### Step 2: Differentiate each term
1. The derivative of \( x^2 \) is:
\[ \frac{d}{dx}(x^2) = 2x \]
2. The derivative of \( y^2 \) using the chain rule is:
\[ \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \]
3. The derivative of \( -3xy \) using the product rule is:
\[ \frac{d}{dx}(-3xy) = -3\left(x \frac{dy}{dx} + y\right) \]
4. The derivative of the constant \( 1 \) is:
\[ \frac{d}{dx}(1) = 0 \]

### Step 3: Combine the derivatives
Putting it all together, we have:
\[ 2x + 2y \frac{dy}{dx} - 3\left(x \frac{dy}{dx} + y\right) + 0 = 0 \]
This simplifies to:
\[ 2x + 2y \frac{dy}{dx} - 3x \frac{dy}{dx} - 3y = 0 \]

### Step 4: Rearrange the equation
Now, we will collect all the terms involving \( \frac{dy}{dx} \) on one side and the constant terms on the other side:
\[ 2y \frac{dy}{dx} - 3x \frac{dy}{dx} = 3y - 2x \]

### Step 5: Factor out \( \frac{dy}{dx} \)
Factoring out \( \frac{dy}{dx} \) gives us:
\[ (2y - 3x) \frac{dy}{dx} = 3y - 2x \]

### Step 6: Solve for \( \frac{dy}{dx} \)
Now, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{3y - 2x}{2y - 3x} \]

### Final Answer
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{3y - 2x}{2y - 3x} \]

---