4x - x²y + y³ = 10\nfind the value of \\frac{dy}{dx} at the point (1, 2).\nchoose 1 answer:\n(a)…

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (4x - x^{2}y + y^{3}) term - by - term. Using the sum/difference rule ((u\pm v\pm w)'=u'\pm v'\pm w'), we have (\frac{d}{dx}(4x)-\frac{d}{dx}(x^{2}y)+\frac{d}{dx}(y^{3})=\frac{d}{dx}(10)). The derivative of (4x) with respect to (x) is (4) (since (\frac{d}{dx}(ax)=a) for (a = 4)). For (\frac{d}{dx}(x^{2}y)), use the product rule ((uv)' = u'v+uv'), where (u = x^{2}), (u'=2x) and (v = y), (v'=\frac{dy}{dx}). So (\frac{d}{dx}(x^{2}y)=2xy+x^{2}\frac{dy}{dx}). For (\frac{d}{dx}(y^{3})), use the chain rule. Let (u = y), then (\frac{d}{dx}(y^{3})=3y^{2}\frac{dy}{dx}). The derivative of a constant (10) with respect to (x) is (0). So, (4-(2xy + x^{2}\frac{dy}{dx})+3y^{2}\frac{dy}{dx}=0).

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

Expand the left - hand side: (4-2xy - x^{2}\frac{dy}{dx}+3y^{2}\frac{dy}{dx}=0). Group the terms with (\frac{dy}{dx}) together: ((3y^{2}-x^{2})\frac{dy}{dx}=2xy - 4). Then (\frac{dy}{dx}=\frac{2xy - 4}{3y^{2}-x^{2}}).

Step3: Substitute (x = 1) and (y = 2)

Substitute (x = 1) and (y = 2) into (\frac{dy}{dx}=\frac{2xy - 4}{3y^{2}-x^{2}}). (\frac{dy}{dx}=\frac{2\times1\times2-4}{3\times2^{2}-1^{2}}=\frac{4 - 4}{12 - 1}=0).

Answer:

D. (0)