Find dy/dx x^3+y^3=6xy-1

Problem

x3+y3=6*x*y−1

Solution

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

d(x3)/d(x)+d(y3)/d(x)=d(6*x*y−1)/d(x)

2. Apply the power rule and product rule to the terms, noting that the derivative of y3 is 3*y2d(y)/d(x) and the derivative of 6*x*y requires the product rule.

3*x2+3*y2d(y)/d(x)=6*y+6*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)−6*xd(y)/d(x)=6*y−3*x2

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

d(y)/d(x)*(3*y2−6*x)=6*y−3*x2

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

d(y)/d(x)=(6*y−3*x2)/(3*y2−6*x)

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

d(y)/d(x)=(2*y−x2)/(y2−2*x)

Final Answer

d(y)/d(x)=(2*y−x2)/(y2−2*x)

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