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

Problem

x2+y2=2*x*y

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the chain rule to y2

d(x2)/d(x)+d(y2)/d(x)=(d(2)*x*y)/d(x)

2. Apply the power rule to the left side and the product rule to the right side.

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

3. Isolate the terms containing d(y)/d(x) on one side of the equation by subtracting 2*xd(y)/d(x) and 2*x from both sides.

2*yd(y)/d(x)−2*xd(y)/d(x)=2*y−2*x

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

d(y)/d(x)*(2*y−2*x)=2*y−2*x

5. Solve for the derivative by dividing both sides by (2*y−2*x) noting that for y≠x the expression simplifies to 1.

d(y)/d(x)=(2*y−2*x)/(2*y−2*x)

6. Simplify the resulting fraction.

d(y)/d(x)=1

Final Answer

d(y)/d(x)=1

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