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

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

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