Find dy/dx x^3+y^3=12xy

Problem

x3+y3=12*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(12)*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)=12*y+12*xd(y)/d(x)

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

3*y2d(y)/d(x)−12*xd(y)/d(x)=12*y−3*x2

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

d(y)/d(x)*(3*y2−12*x)=12*y−3*x2

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

d(y)/d(x)=(12*y−3*x2)/(3*y2−12*x)

6. Simplify the fraction by dividing the numerator and the denominator by the greatest common factor, which is 3

d(y)/d(x)=(4*y−x2)/(y2−4*x)

Final Answer

d(y)/d(x)=(4*y−x2)/(y2−4*x)

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