Find dy/dx x^3+y^3=3xy

Problem

x3+y3=3*x*y

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the chain rule to the y3 term.

d(x3)/d(x)+d(y3)/d(x)=(d(3)*x*y)/d(x)

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

3*x2+3*y2d(y)/d(x)=3*y+3*xd(y)/d(x)

3. Divide by 3 to simplify the equation.

x2+y2d(y)/d(x)=y+xd(y)/d(x)

4. Group the terms containing d(y)/d(x) on one side and the remaining terms on the other side.

y2d(y)/d(x)−xd(y)/d(x)=y−x2

5. Factor out d(y)/d(x) from the left side.

d(y)/d(x)*(y2−x)=y−x2

6. Solve for dy/dx by dividing both sides by the factor (y2−x)

d(y)/d(x)=(y−x2)/(y2−x)

Final Answer

d(y)/d(x)=(y−x2)/(y2−x)

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