(c) Find frac{d y}{d x} where x+y+sqrt{x y}=1.

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Step:1 ：Understand the problem: We are asked to find the derivative of y with respect to x, given the equation (x + y + sqrt{xy} = 1).Step:2 ：Differentiate both sides of the equation with respect to x: (1 + frac{dy}{dx} + frac{1}{2} cdot frac{1}{sqrt{xy}} cdot (y + x cdot frac{dy}{dx}) = 0).Step:3 ：Simplify the equation by multiplying through by (2sqrt{xy}) to clear the fraction: (2sqrt{xy} + 2sqrt{xy} cdot frac{dy}{dx} + y + x cdot frac{dy}{dx} = 0).Step:4 ：Collect terms involving (frac{dy}{dx}): ((2sqrt{xy} + x) cdot frac{dy}{dx} = -2sqrt{xy} - y).Step:5 ：Solve for (frac{dy}{dx}): (frac{dy}{dx} = frac{-2sqrt{xy} - y}{2sqrt{xy} + x}).Step:6 ：Check the result by substituting (frac{dy}{dx}) back into the differentiated equation to check if it satisfies the equation. If it does, then the result is correct.Step:7 ：(boxed{frac{dy}{dx} = frac{-2sqrt{xy} - y}{2sqrt{xy} + x}})

Answer

Step: 1 ：Understand the problem: We are asked to find the derivative of y with respect to x, given the equation (x + y + sqrt{xy} = 1).Step: 2 ：Differentiate both sides of the equation with respect to x: (1 + frac{dy}{dx} + frac{1}{2} cdot frac{1}{sqrt{xy}} cdot (y + x cdot frac{dy}{dx}) = 0).Step: 3 ：Simplify the equation by multiplying through by (2sqrt{xy}) to clear the fraction: (2sqrt{xy} + 2sqrt{xy} cdot frac{dy}{dx} + y + x cdot frac{dy}{dx} = 0).Step: 4 ：Collect terms involving (frac{dy}{dx}): ((2sqrt{xy} + x) cdot frac{dy}{dx} = -2sqrt{xy} - y).Step: 5 ：Solve for (frac{dy}{dx}): (frac{dy}{dx} = frac{-2sqrt{xy} - y}{2sqrt{xy} + x}).Step: 6 ：Check the result by substituting (frac{dy}{dx}) back into the differentiated equation to check if it satisfies the equation. If it does, then the result is correct.Step: 7 ：(boxed{frac{dy}{dx} = frac{-2sqrt{xy} - y}{2sqrt{xy} + x}})

(c) Find $\frac{d y}{d x}$ where $x + y + \sqrt{x y} = 1$ .

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Popular Problems

(c) Find dydx where x+y+xy=1.

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Popular Problems

(c) Find $\frac{d y}{d x}$ where $x + y + \sqrt{x y} = 1$ .