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

Answer

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

### Step-by-Step Solution:

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

2. **Differentiate both sides with respect to \( x \):**
- The left side:
- The derivative of \( x^2 \) is \( 2x \) (using the power rule).
- The derivative of \( -x \) is \( -1 \).
- Therefore, the derivative of the left side is:
\[ \frac{d}{dx}(x^2 - x) = 2x - 1 \]

- The right side:
- The derivative of \( 2y \) is \( 2 \frac{dy}{dx} \) (using the chain rule, since \( y \) is a function of \( x \)).
- Therefore, the derivative of the right side is:
\[ \frac{d}{dx}(2y) = 2 \frac{dy}{dx} \]

3. **Set the derivatives equal to each other:**
\[ 2x - 1 = 2 \frac{dy}{dx} \]

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

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