2y^{2}-x^{2}+x^{3}y = 2\nfind \\frac{dy}{dx}.\nchoose 1 answer:\na \\frac{4y + x^{3}}{2x - 3x^{2}y}\nb…

Explanation:

Step1: Differentiate both sides

Differentiate (2y^{2}-x^{2}+x^{3}y = 2) with respect to (x). Using the chain - rule ((u^{n})^\prime=nu^{n - 1}u^\prime) for (2y^{2}) (where (u = y)), the power - rule ((x^{n})^\prime=nx^{n-1}) for (-x^{2}), and the product - rule ((uv)^\prime = u^\prime v+uv^\prime) for (x^{3}y) (where (u=x^{3}), (v = y)). (\frac{d}{dx}(2y^{2})-\frac{d}{dx}(x^{2})+\frac{d}{dx}(x^{3}y)=\frac{d}{dx}(2)) (4y\frac{dy}{dx}-2x+(3x^{2}y+x^{3}\frac{dy}{dx}) = 0)

Step2: Solve for (\frac{dy}{dx})

Group the terms with (\frac{dy}{dx}) on one side: (4y\frac{dy}{dx}+x^{3}\frac{dy}{dx}=2x - 3x^{2}y) Factor out (\frac{dy}{dx}): (\frac{dy}{dx}(4y + x^{3})=2x-3x^{2}y) Then (\frac{dy}{dx}=\frac{2x - 3x^{2}y}{4y+x^{3}}) (by dividing both sides by (4y + x^{3}))

Answer:

None of the options A, B, C are correct. The correct derivative (\frac{dy}{dx}=\frac{2x - 3x^{2}y}{4y + x^{3}})