Find dy/dx 2xy+y^2=x+y

Problem

2*x*y+y2=x+y

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the chain rule where necessary.

d()/d(x)*(2*x*y+y2)=d()/d(x)*(x+y)

2. Apply the product rule to the term 2*x*y and the power rule with the chain rule to y2

2*y+2*xd(y)/d(x)+2*yd(y)/d(x)=1+d(y)/d(x)

3. Group all terms containing d(y)/d(x) on one side of the equation and move all other terms to the opposite side.

2*xd(y)/d(x)+2*yd(y)/d(x)−d(y)/d(x)=1−2*y

4. Factor out the common factor d(y)/d(x) from the left side.

d(y)/d(x)*(2*x+2*y−1)=1−2*y

5. Solve for the derivative by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(1−2*y)/(2*x+2*y−1)

Final Answer

d(y)/d(x)=(1−2*y)/(2*x+2*y−1)

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