homework7: problem 5\n(2 points)\nfind \\( \\frac{d y}{d x} \\) by implicit differentiation.\n\\( 3…
homework7: problem 5\n(2 points)\nfind \\( \\frac{d y}{d x} \\) by implicit differentiation.\n\\( 3 x^{3}+x^{2} y-x y^{3}=-2 \\)\nanswer: \\( \\frac{d y}{d x}= \\)\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have unlimited attempts remaining.\nemail instructor\npage generated october 19, 2025, 8:39:39 pm cdt\nwebwork \\( \\odot 1996-2024 \\) | theme: math4_ttu | ww_version: 2.19 | pg_version 2.1\nthe webwork project
Answer
Explanation:
Step1: Differentiate each term
Differentiate (3x^{3}+x^{2}y - xy^{3}=-2) term - by - term with respect to (x). Using the power rule (\frac{d}{dx}(x^{n})=nx^{n - 1}), the product rule (\frac{d}{dx}(uv)=u'v + uv') (where (u) and (v) are functions of (x)), and the chain rule (\frac{d}{dx}(y^{n})=ny^{n - 1}\frac{dy}{dx}) (since (y) is a function of (x)).
- (\frac{d}{dx}(3x^{3})=9x^{2})
- For (\frac{d}{dx}(x^{2}y)), let (u = x^{2}) and (v = y). Then (\frac{d}{dx}(x^{2}y)=2xy+x^{2}\frac{dy}{dx})
- For (\frac{d}{dx}(xy^{3})), let (u = x) and (v = y^{3}). Then (\frac{d}{dx}(xy^{3})=y^{3}+3xy^{2}\frac{dy}{dx})
- (\frac{d}{dx}(-2)=0)
The differentiated equation is: (9x^{2}+2xy + x^{2}\frac{dy}{dx}-y^{3}-3xy^{2}\frac{dy}{dx}=0)
Step2: Solve for (\frac{dy}{dx})
Group the terms with (\frac{dy}{dx}) on one side: (x^{2}\frac{dy}{dx}-3xy^{2}\frac{dy}{dx}=y^{3}-9x^{2}-2xy) Factor out (\frac{dy}{dx}): (\frac{dy}{dx}(x^{2}-3xy^{2})=y^{3}-9x^{2}-2xy) Then (\frac{dy}{dx}=\frac{y^{3}-9x^{2}-2xy}{x^{2}-3xy^{2}})
Answer:
(\frac{y^{3}-9x^{2}-2xy}{x^{2}-3xy^{2}})