Find \( d y / d x \) \( 2 x^{3}+5 x y=y^{2} \) Type answer \( y^{\prime}= \) (numerator)/(denominator)

To find \( \frac{dy}{dx} \) for the equation \( 2x^3 + 5xy = y^2 \), we can use implicit differentiation. Starting with the equation, we differentiate both sides with respect to \( x \): 1. Differentiate \( 2x^3 \) to get \( 6x^2 \). 2. For \( 5xy \), we apply the product rule: \( 5 \left( y + x \frac{dy}{dx} \right) \). 3. For \( y^2 \), we use the chain rule to get \( 2y \frac{dy}{dx} \). Putting it all together: \[ 6x^2 + 5 \left( y + x \frac{dy}{dx} \right) = 2y \frac{dy}{dx} \] Now, rearranging terms gives us: \[ 6x^2 + 5y = 2y \frac{dy}{dx} - 5x \frac{dy}{dx} \] Factoring out \( \frac{dy}{dx} \) from the right-hand side: \[ 6x^2 + 5y = \frac{dy}{dx}(2y - 5x) \] Finally, we can isolate \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{6x^2 + 5y}{2y - 5x} \] Thus, the final answer is: \( y' = \frac{6x^2 + 5y}{2y - 5x} \)