2x^{3}-5xy - y^{2}=3\nfind \\frac{dy}{dx}.\nchoose 1 answer:\na \\frac{6x^{2}-5y}{5x + 2y}\nb…

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (2x^{3}-5xy - y^{2}=3) term - by - term. Using the power rule (\frac{d}{dx}(x^{n})=nx^{n - 1}), (\frac{d}{dx}(2x^{3})=6x^{2}). For the term (-5xy), use the product rule (\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}), where (u=-5x) and (v = y). So (\frac{d}{dx}(-5xy)=-5y-5x\frac{dy}{dx}). For the term (-y^{2}), use the chain rule (\frac{d}{dx}(u^{n})=nu^{n - 1}\frac{du}{dx}), where (u = y) and (n = 2). So (\frac{d}{dx}(-y^{2})=-2y\frac{dy}{dx}). Differentiating the right - hand side (\frac{d}{dx}(3)=0). The derivative of the left - hand side is (6x^{2}-5y-5x\frac{dy}{dx}-2y\frac{dy}{dx}), and the derivative of the right - hand side is (0). So (6x^{2}-5y-5x\frac{dy}{dx}-2y\frac{dy}{dx}=0).

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

Group the terms with (\frac{dy}{dx}) together: (-5x\frac{dy}{dx}-2y\frac{dy}{dx}=5y - 6x^{2}). Factor out (\frac{dy}{dx}): (\frac{dy}{dx}(-5x - 2y)=5y - 6x^{2}). Then (\frac{dy}{dx}=\frac{6x^{2}-5y}{5x + 2y}).

Answer:

A. (\frac{6x^{2}-5y}{5x + 2y})