Find `dy/dx if x-2y=x^2`

Answer

To find \( \frac{dy}{dx} \) for the equation \( x - 2y = x^2 \), we will differentiate both sides with respect to \( x \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x - 2y = x^2 \]

2. **Differentiate both sides with respect to \( x \):**
- The derivative of \( x \) with respect to \( x \) is \( 1 \).
- For \( -2y \), we apply the product rule. The derivative is \( -2 \frac{dy}{dx} \).
- The derivative of \( x^2 \) is \( 2x \) (using the power rule).

So, differentiating the equation gives:
\[ 1 - 2 \frac{dy}{dx} = 2x \]

3. **Rearrange the equation to solve for \( \frac{dy}{dx} \):**
\[ -2 \frac{dy}{dx} = 2x - 1 \]

4. **Divide both sides by -2 to isolate \( \frac{dy}{dx} \):**
\[ \frac{dy}{dx} = \frac{1 - 2x}{2} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{1 - 2x}{2} \]