if (2xy^{2}-3x^{2}y = 6x), then (\frac{dy}{dx}=) \na) (\frac{6}{1y - 6x}) \nb) (\frac{2y^{2}-2xy…

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (2xy^{2}-3x^{2}y = 6x) using product - rule ((uv)^\prime=u^\prime v + uv^\prime). For the left - hand side: The derivative of (2xy^{2}) is (2y^{2}+4xy\frac{dy}{dx}) (using product rule where (u = 2x), (v = y^{2}), (u^\prime=2), (v^\prime = 2y\frac{dy}{dx})). The derivative of (-3x^{2}y) is (-6xy-3x^{2}\frac{dy}{dx}) (using product rule where (u=-3x^{2}), (v = y), (u^\prime=-6x), (v^\prime=\frac{dy}{dx})). The derivative of the right - hand side (6x) is (6). So, (2y^{2}+4xy\frac{dy}{dx}-6xy - 3x^{2}\frac{dy}{dx}=6).

Step2: Isolate (\frac{dy}{dx})

Group the terms with (\frac{dy}{dx}) on one side: (4xy\frac{dy}{dx}-3x^{2}\frac{dy}{dx}=6 + 6xy-2y^{2}). Factor out (\frac{dy}{dx}): (\frac{dy}{dx}(4xy - 3x^{2})=6 + 6xy-2y^{2}). Then (\frac{dy}{dx}=\frac{-2y^{2}+6xy + 6}{4xy-3x^{2}}).

Answer:

D. (\frac{-2y^{2}+6xy + 6}{4xy-3x^{2}})