Find dy/dx x^3+y^3-6xy=0

Problem

x^3+y^3−6*x*y=0

Solution

1. Differentiate both sides of the equation with respect to x treating y as a function of x and applying the chain rule to the y^3 term.

d(x^3)/d(x)+d(y^3)/d(x)−(d(6)*x*y)/d(x)=d(0)/d(x)

2. Apply the power rule and the product rule to the terms.

3*x^2+3*y^2d(y)/d(x)−(6*y+6*xd(y)/d(x))=0

3. Distribute the negative sign and group all terms containing d(y)/d(x) on one side of the equation.

3*x^2+3*y^2d(y)/d(x)−6*y−6*xd(y)/d(x)=0

3*y^2d(y)/d(x)−6*xd(y)/d(x)=6*y−3*x^2

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

d(y)/d(x)*(3*y^2−6*x)=6*y−3*x^2

5. Solve for d(y)/d(x) by dividing both sides by (3*y^2−6*x)

d(y)/d(x)=(6*y−3*x^2)/(3*y^2−6*x)

6. Simplify the fraction by dividing the numerator and the denominator by 3

d(y)/d(x)=(2*y−x^2)/(y^2−2*x)

Final Answer

d(y)/d(x)=(2*y−x^2)/(y^2−2*x)

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