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Question 33\newlineTips\newlinesmartsolve\newline77.77pts\newlineIf \newlinex3+3xy+2y3=17x^{3}+3xy+2y^{3}=17, then in terms of \newlinexx and \newlineyy,(dy)/(dx)=(dy)/(dx)=\newline-((x2+y)/(x+2y2))((x^{2}+y)/(x+2y^{2}))\newline-((x2+y)/(x+2y))((x^{2}+y)/(x+2y))\newline-((x2+y)/(2y2))((x^{2}+y)/(2y^{2}))\newline-((x2+y)/(x+y2))((x^{2}+y)/(x+y^{2}))

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Q. Question 33\newlineTips\newlinesmartsolve\newline77.77pts\newlineIf \newlinex3+3xy+2y3=17x^{3}+3xy+2y^{3}=17, then in terms of \newlinexx and \newlineyy,(dy)/(dx)=(dy)/(dx)=\newline-((x2+y)/(x+2y2))((x^{2}+y)/(x+2y^{2}))\newline-((x2+y)/(x+2y))((x^{2}+y)/(x+2y))\newline-((x2+y)/(2y2))((x^{2}+y)/(2y^{2}))\newline-((x2+y)/(x+y2))((x^{2}+y)/(x+y^{2}))
  1. Question Prompt: Question Prompt: If x3+3xy+2y3=17x^3 + 3xy + 2y^3 = 17, find dydx\frac{dy}{dx} in terms of xx and yy.
  2. Implicit Differentiation: Use implicit differentiation on both sides of the equation with respect to xx. Differentiate each term separately:\newline- Derivative of x3x^3 is 3x23x^2.\newline- Derivative of 3xy3xy using the product rule is 3xdydx+3y3x \frac{dy}{dx} + 3y.\newline- Derivative of 2y32y^3 using the chain rule is 6y2dydx6y^2 \frac{dy}{dx}.\newline- The derivative of the constant 1717 is 00.
  3. Combine Derivatives: Combine the derivatives:\newline3x2+3xdydx+3y+6y2dydx=0 3x^2 + 3x \frac{dy}{dx} + 3y + 6y^2 \frac{dy}{dx} = 0
  4. Solve for dy/dx: Rearrange to solve for dydx\frac{dy}{dx}:\newline3xdydx+6y2dydx=3x23y 3x \frac{dy}{dx} + 6y^2 \frac{dy}{dx} = -3x^2 - 3y \newline(3x+6y2)dydx=3x23y (3x + 6y^2) \frac{dy}{dx} = -3x^2 - 3y \newlinedydx=3x23y3x+6y2 \frac{dy}{dx} = \frac{-3x^2 - 3y}{3x + 6y^2} \newlinedydx=(x2+y)x+2y2 \frac{dy}{dx} = \frac{-(x^2 + y)}{x + 2y^2}

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