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(x+y)2 * d2y/dx2 = xy + 1. Differentiate both sides in 2(x+y).

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Answer

dydx=−2x−yx+2y\frac{dy}{dx} = \frac{-2x-y}{x+2y}

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3. Find dydx\frac{dy}{dx} for (x+y)2=xy+1(x+y)^2 = xy+1

The student's work correctly differentiated both sides of the equation:

2(x+y)(1+dydx)=y+xdydx2(x+y)\left(1 + \frac{dy}{dx}\right) = y + x\frac{dy}{dx}

  1. Expand the left side of the equation.

    2(x+y)+2(x+y)dydx=y+xdydx2(x+y) + 2(x+y)\frac{dy}{dx} = y + x\frac{dy}{dx} 2x+2y+(2x+2y)dydx=y+xdydx2x + 2y + (2x+2y)\frac{dy}{dx} = y + x\frac{dy}{dx}

  2. Group terms containing dydx\frac{dy}{dx} on one side and other terms on the other side.

    (2x+2y)dydx−xdydx=y−2x−2y(2x+2y)\frac{dy}{dx} - x\frac{dy}{dx} = y - 2x - 2y

  3. Factor out dydx\frac{dy}{dx} and simplify the terms inside the parentheses.

    dydx(2x+2y−x)=−2x−y\frac{dy}{dx}(2x+2y-x) = -2x - y dydx(x+2y)=−2x−y\frac{dy}{dx}(x+2y) = -2x - y

  4. Isolate dydx\frac{dy}{dx}.

    dydx=−2x−yx+2y\frac{dy}{dx} = \frac{-2x-y}{x+2y}
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